#prove sin 2x | Explore Tumblr Posts and Blogs

cergisoft · 22 days

Text

What is Tan2x in Trigonometry

What is Tan2x in Trigonometry?

In trigonometry, the term Tan2x refers to the tangent of double angles. It’s a key concept in trigonometric identities and equations, playing a crucial role in simplifying complex trigonometric expressions and solving problems involving angles. This article will explore What is Tan2x in Trigonometry, its significance, and its applications in trigonometry.

Understanding Tan2x

The Tan2x Formula

The formula for Tan2x is a result of trigonometric identities. The double-angle formula for tangent is:

tan⁡(2x)=2tan⁡(x)1−tan⁡2(x)\tan(2x) = \frac{2\tan(x)}{1 — \tan²(x)}tan(2x)=1−tan2(x)2tan(x)

This formula allows you to find the tangent of twice an angle using the tangent of the original angle. It simplifies calculations and is useful in various trigonometric proofs and applications.

Applications of Tan2x

Solving Trigonometric Equations: The Tan2x formula is often used to solve trigonometric equations where angles are doubled. It helps in simplifying and finding the values of angles that satisfy the equation.

Graphing Trigonometric Functions: Understanding the behavior of tan2x is crucial when graphing tangent functions. The formula aids in determining the shape and characteristics of the graph.

Trigonometric Proofs: The Tan2x formula is used in proving various trigonometric identities and equations. It is an essential tool for mathematicians and students in solving complex problems.

Example Problem

Let’s consider an example to illustrate the use of the Tan2x formula. Suppose we want to find the tangent of twice an angle where tan⁡(x)=1\tan(x) = 1tan(x)=1.

Using the formula:

tan⁡(2x)=2tan⁡(x)1−tan⁡2(x)\tan(2x) = \frac{2\tan(x)}{1 — \tan²(x)}tan(2x)=1−tan2(x)2tan(x)

Substitute tan⁡(x)=1\tan(x) = 1tan(x)=1:

tan⁡(2x)=2×11−12\tan(2x) = \frac{2 \times 1}{1–1²}tan(2x)=1−122×1 tan⁡(2x)=20\tan(2x) = \frac{2}{0}tan(2x)=02

Since dividing by zero is undefined, this indicates that tan⁡(2x)\tan(2x)tan(2x) is undefined, which corresponds to the fact that the tangent function has vertical asymptotes at these points.

FAQs

1. What is the significance of the Tan2x formula in trigonometry?

The Tan2x formula simplifies the calculation of the tangent of double angles, making it easier to solve trigonometric equations and proofs.

2. How do I derive the Tan2x formula?

The Tan2x formula is derived from the tangent addition formula. By applying the formula for tan⁡(x+x)\tan(x + x)tan(x+x), you can derive the expression 2tan⁡(x)1−tan⁡2(x)\frac{2\tan(x)}{1 — \tan²(x)}1−tan2(x)2tan(x).

3. Can the Tan2x formula be used for any angle?

Yes, the Tan2x formula can be used for any angle where the tangent function is defined. However, it is important to note that it becomes undefined where tan⁡(x)=±1\tan(x) = \pm 1tan(x)=±1, as the denominator becomes zero.

4. How does Tan2x affect the graph of the tangent function?

The graph of tan2x will have its period compressed compared to the standard tangent function. This is due to the doubling of the angle, which affects the frequency of the function’s oscillation.

Additional Insights on Tan2x

1. Derivation of the Tan2x Formula

To derive the formula for Tan2x, we start with the tangent addition formula:

tan⁡(A+B)=tan⁡(A)+tan⁡(B)1−tan⁡(A)tan⁡(B)\tan(A + B) = \frac{\tan(A) + \tan(B)}{1 — \tan(A)\tan(B)}tan(A+B)=1−tan(A)tan(B)tan(A)+tan(B)

For the case where A=B=xA = B = xA=B=x, the formula becomes:

tan⁡(2x)=tan⁡(x)+tan⁡(x)1−tan⁡(x)tan⁡(x)\tan(2x) = \frac{\tan(x) + \tan(x)}{1 — \tan(x)\tan(x)}tan(2x)=1−tan(x)tan(x)tan(x)+tan(x)

Simplify the expression:

tan⁡(2x)=2tan⁡(x)1−tan⁡2(x)\tan(2x) = \frac{2\tan(x)}{1 — \tan²(x)}tan(2x)=1−tan2(x)2tan(x)

This derivation shows how the Tan2x formula is obtained from the fundamental tangent addition formula, providing a clear understanding of its application.

2. Comparing Tan2x with Other Double-Angle Formulas

In trigonometry, double-angle formulas are used for sine, cosine, and tangent functions. Here’s a quick comparison:

Sine Double-Angle Formula: sin⁡(2x)=2sin⁡(x)cos⁡(x)\sin(2x) = 2\sin(x)\cos(x)sin(2x)=2sin(x)cos(x)

Cosine Double-Angle Formula: cos⁡(2x)=cos⁡2(x)−sin⁡2(x)\cos(2x) = \cos²(x) — \sin²(x)cos(2x)=cos2(x)−sin2(x) or cos⁡(2x)=2cos⁡2(x)−1\cos(2x) = 2\cos²(x) — 1cos(2x)=2cos2(x)−1 or cos⁡(2x)=1−2sin⁡2(x)\cos(2x) = 1–2\sin²(x)cos(2x)=1−2sin2(x)

Tangent Double-Angle Formula: tan⁡(2x)=2tan⁡(x)1−tan⁡2(x)\tan(2x) = \frac{2\tan(x)}{1 — \tan²(x)}tan(2x)=1−tan2(x)2tan(x)

Each formula serves a unique purpose and is used based on the trigonometric function involved. Understanding these formulas helps in solving complex problems and in deriving other identities.

3. Practical Applications of Tan2x

Engineering and Physics: In engineering and physics, the Tan2x formula is used in various applications, including wave analysis and signal processing, where understanding the behavior of trigonometric functions is crucial.

Computer Graphics: In computer graphics, trigonometric functions are used for transformations, rotations, and rendering. The Tan2x formula helps in accurately representing these transformations.

Astronomy: The formula can be applied in astronomy for calculating angles and distances in celestial observations, where precise trigonometric calculations are required.

4. Common Mistakes and Misconceptions

Ignoring Domain Restrictions: One common mistake is ignoring the domain restrictions where the formula becomes undefined. It’s essential to consider these restrictions while solving problems.

Misapplying the Formula: Ensure that you are correctly applying the Tan2x formula. Misapplying it can lead to incorrect results. Double-check the angle and its tangent value before using the formula.

FAQs:

5. How can I use the Tan2x formula in real-life situations?

The Tan2x formula can be applied in fields such as engineering, computer graphics, and astronomy, where precise trigonometric calculations are required for various applications.

6. Are there any visual tools to help understand Tan2x?

Yes, graphing calculators and software can visually represent the behavior of the Tan2x function. These tools help in understanding the function’s characteristics and how it compares to the standard tangent function.

7. What should I do if I encounter an undefined value while using the Tan2x formula?

If the formula results in an undefined value, it typically means that the tangent of the angle is such that the denominator becomes zero. In such cases, review the problem context and consider the periodic nature of the tangent function to interpret the results correctly.

8. Can I apply the Tan2x formula to angles larger than 360 degrees?

Yes, the formula can be applied to angles larger than 360 degrees by considering the periodic nature of the tangent function. The tangent function repeats every 180 degrees, so angles can be reduced to their corresponding values within one period.

Conclusion

The Tan2x formula is a fundamental tool in trigonometry that simplifies the calculation of the tangent of double angles. Its derivation from the tangent addition formula and its various applications make it a critical concept for students and professionals alike. By understanding and applying the Tan2x formula correctly, you can solve complex trigonometric problems with ease.

#Tan2x Formula

0 notes

pint4punt · 8 months

Text

And the Lombardi goes to:

What NFL Fanbases’ Bullshit Sob Story Deserves it All

Championship Sunday is finally upon us and boy have the scriptwriters got compelling stories this go round! With the playoffs narrowed down to the final four, we’ve got unforgettable narratives for…….almost all of the QBs still in this! Statistics, scheme, and matchups be damned (let’s be honest you’ve heard every prediction imaginable since these matchups were set).

Today we’ll be focusing on what really matters. Who has the most compelling sob story for quality Super Bowl ratings! Who could forget classics such as ‘Tom Brady Builds a Super Team to Prove it was Him All Along’ or the all star cast of Matthew Stafford, Odell Beckham Jr., and Von Miller in ‘The NFL Really Wants the Robust LA Market!’ featuring Snoop Dogg on the original motion picture soundtrack! You see a ‘may the best team win approach’ is fine for a place setter, but storylines put asses in the seats! And don’t worry, as bad as our jokes are, Jo Koy had little to no involvement in this article! So which Lombardi win makes the best headline? Here are the nominees:

Lamar Jackson

Since he declared for the draft Lamar Jackson has had to overcome widespread perception that he was closer to a skill position player than a signal caller. Many scouts questioned his ability to cut it in the pros at the Quarterback position. This stigma was so strong that Lamar fell all the way to the last pick of the First Round.

John Harbaugh believed in Lamar so strongly, that he reinvented himself and rebuilt his entire Offense around Jackson’s skill set. Shit, I would roll the dice too if my job was on the line after several middling seasons with an aging Joe Flacco! Either it works and you look like a genius or it fails catastrophically and you go out with a bang. Luckily for Harbaugh and Lamar, it was the latter with the now presumably 2x MVP set to play in his first AFC Championship game. Lamar has already shaken off the narrative of him not showing up in the playoffs just by making this game. But knocking off the QB who owns a 3-1 record over him for a trip to Sin City would be Oscar gold! Not bad for a running back.

Jared Goff

Everyone loves a comeback story. An underdog achieving in the face of adversity. A character finding a new lease on life and showing all the doubters they were wrong about him. There may not be a better archetype of that narrative than former 1st overall pick Jared Goff and the historically downtrodden Detroit Lions.

After being sent to die in Detroit, a franchise that has seemingly been in a perpetual state of rebuild for damn near the entirety of its existence, Goff watched from afar as Sean McVay gushed over his new QB Matthew Stafford and his old team held up the trophy he could never give them. Following two seasons with no playoff appearances, Goff seized the opportunity to prove to McVay that he’s not a virgin anymore by out-dueling his replacement to give Detroit their first playoff win in 30 years and keep Stafford winless in Ford Field in the post season.

Now he has the Lions on the precipice of a Super Bowl appearance with a chance to give Detroit its first ever Lombardi. Apparently the grass isn’t always greener on the other side……except in this case where it totally is seeing as how the Rams already won their second Super Bowl like two seasons ago thanks specifically to the Stafford trade, while the Lions are still fighting the good fight for Lombardi number one. But hey, everyone’s happy now right?

Brock Purdy

Is there a more inspiring yet polarizing figure in the NFL than Brock Purdy? He’s like a ‘Make a Wish’ kid that the media wants to beat the living shit out of on sight. In fact, we’re fairly certain Ryan Clark has his picture on a punching bag in his home and regularly unleashes a fury of blows onto an 8.5 x 11 cut out of Mr. Irrelevant. But why? Why does the NFL version of Michael Cera incite as much vitriol as….well Michael Cera?

As the last pick of the draft, Purdy was lucky to even make the final roster of a team that was coming off its 2nd conference championship appearance in 3 years. The depth chart in front of him featured a veteran who had taken the Niners to a Super Bowl and a guy Shanahan gave up his left nut for the year before. But Purdy was like that one sperm that ultimately reached conception, despite overwhelming odds, he just wouldn’t be denied!

Following season ending injuries to both Trey Lance and Jimmy G, Purdy took over a Super Bowl contender mid season…..and might have led them to a Super Bowl were it not for a devastating elbow injury in the NFC Championship that ended San Francisco’s dream run. One year and one UCL repair later, Purdy and the Niners picked right up where they left off and now find themselves back in the Conference Title game as if nothing happened. That’s like Greta Thunberg single-handedly fighting off an army of Exxon Oil’s greatest assassins or Joe Biden finishing a Dr. Seuss book with no assistance. We don’t know what’s up the media’s ass, but that seems pretty impressive for a game manager!

Patrick Mahomes

And then there’s Mahomes. The Meryl Streep in all of this. The one who had the least impressive performance with the least impressive story who you just know is going to spoil all the fun of the more deserving nominees. On paper, this fucker shouldn’t even be here let alone have a puncher’s chance with the way their regular season looked. But that’s what makes Patrick Mahomes, well Patrick Mahomes.

He’s like the Lebron James of football. Or better yet, the Tom Brady….of football…. Poor analogies aside, Mahomes defies all logic and drags inferior teams to the promise land. Sure the script doesn’t favor him this year, but guess what? It didn’t favor him last week against the Buffalo team that finally had him their house this postseason! Or in the Super Bowl last year against Jalen Hurts who mostly outplayed him before…..whatever this season was!

The simple fact of the matter is anyone who pays attention already knows how this ends. Mahomes is going to win the god damn Super Bowl….again. Why?!!! Because the academy, the scriptwriters, and the NFL writ large are lazy as all hell! And dynasties keep the fans coming back, just on the off chance that their pessimistic hopes of a downfall might finally come to fruition….even though they probably won’t until the NFL’s new golden boy is well past his prime. And the Lombardi goes to…….the same son of a bitch it always goes to!!!!! Hopefully we’re wrong and we get a fresh story this time around. Otherwise tune in next year for another crappy sequel with a few minor tweaks. Happy Championship Sunday!

#championshipsunday #nfl #afc championship #nfcchampionship #kansas city chiefs #baltimoreravens #detroit lions #san fransisco 49ers #football

1 note · View note

fromanartistsheart2 · 11 months

Text

I had a hard time

Writing this rhyme

Deciding which stories

To share that aren't gory

2x brain surgery survivor

B4 the age of 2

Family was worried

About what I could do

3 more surgeries

For an ongoing health condition

That doesn't have a cure

Of that, Drs r sure

Raised by a single daddy

Mother nowhere to be found

We heard nothing for years

Not a single sound

My mom came along

When I was just 12

And she is the bomb

About it I wrote a song

Engaged at 19

To a man that was obscene

Going to strip bars

When I would ask where u r

I caught him cheating

In a private room

And that was the end of

The relationship that was doomed

From Carolina

Down n the south

And to go with it

I have a smart mouth

Met my next migraine

Thru their ex

It was insane

That should have been my sign

At age 21, married then pregnant

I was stuck with him

Didn't know where to go

But it was easy to leave, people told me so

First child born with health issues

That I didn't have enuff tissues

For all my tears

Throughout the years

I was wrong to make

My child behave

And to my husband

Was expected to b a slave

After 6 years

Baby 2 came along

I stayed home with the kids

I thought he was working, but I was so wrong

He lied to me again

And expected me to give in

To join in with his sin

And lie to people again

Begging for food

Which I thought was rude

Begging for rides

Or he would have my hide

Abuse and tyranny

The whole time

It even inspired

Lots of rhymes

Met my new husband

Right here on Facebook

Writing my rhymes

Of the terrible times

Baby #3

And oh woe was me

The oldest out of control

And so my temper rolled

Family interfered

And told my child to disobey

My child was on the losing end

Now matter which way

He listens to the people

Who claim to go to the steeple

But expect him to b a sheeple

And listen to the wrong people

I was suppose to let

His behavior slide

And let him do what he wants

Because to him people lied

So in 2015

When things were past obscene

I met a new guy

And to me he didn't lie

He also wrote rhymes

And I responded most of the time

He was there for me

And my kids 3

So we wrote forth and back

And he never gave me slack

Told me like it was

And never lied because

He is a real man

And he came up with a plan

To rescue me

And my babies, 3

Over time we grew

And by now each other knew

That we loved the other

And had to b with one another

So n 2017

My son made accusations (obscene)

So the cops said we had to leave

And no longer have to cleave

The cops never investigated

Never tried, and it's them I hated

He has big connections

But none that I can prove, just suspections

He got the kids back

And kept them n slack

And now their grades they're failing

Like everyone that I've been telling

Everyday I live

It is for my kids

To save them from him

And everything he's did

He's tried to get me to stop the fight

So he can do everything not right

So that I won't report

And to threats he won't have to resort

So f*cked up

He has them

From what

He's done to them

P*ssed off?!

I'm way beyond

And of the court system

I am not fond

They tell me that my kids

R f*cked up beyond repair

And so I just should stop

And not even care

I should just go on

Live my life and forget

But if I do

I'll have lots of regrets

I tell u all these things

Cause it's for my kids that I breathe

And all the things my ex has done

It just makes me seethe

Fill up with anger

To the point of danger

And go full beast mode

And just explode

But if I do that

He will get what he wants

To prove I'm mentally unstable

And get the kids? I won't

Protect my kids?

The system won't

And my ex?!

I KNOW he won't!

So my heart breaks

With everything he takes

Away from my kids

Just like to him, his family did

Heartwarming Hymns

0 notes

devphilamaths · 3 years

Link

How to prove sin 2x = 2 sin x cos x | Proofs of Trigonometric Identities sin 2x = 2sin x cos x

12th math

IIT JEE

12th class

#prove sin 2x #prove sin 2x = 2 sin x cos x #sin 2x = 2 sin x cos x #Proofs of Trigonometric Identities sin 2x #Proofs of Trigonometric Identities #sin 2x #sin

0 notes

hanawrites404 · 4 years

Text

Twice My Age

Show : Jojo's Bizarre Adventure/Jojo No Kimou Na Bouken

Pairing : Noriko (female Noriaki)/Jotaro Kujo

Warnings : Age-gap romance and sex along with swearing.

Characters : Jotaro Kujo, Noriko (female Noriaki), Jolyne Cujoh

Timeline : Pre-Stone Ocean

This story is based on this song :

This story is also based on the author's headcanon when she had finished reading Stone Ocean and was recovering her brain cells.

Third Person POV

It was a normal afternoon at Florida. Today was Friday, which meant the last day of the working week. Students at the school were frustratingly tapping the heels of their feet on the floor waiting for the school bell to ring and mark the end of the tiring session.

The teacher obviously minded his own business and lectured the pupils without averting his gaze from the blackboard. He kept ignoring the sounds of whispers and paper-tearing and snickers of the children, until for once he recognised one of the students who was continuously talking in his class, and decided to take some action.

"Ms. Cujoh" the teacher called out harshly as he teared his gaze from the blackboard and stared at the student.

The talkative student looked at the teacher and went silent. She had a piece of folded paper in her hand, her arm stretched out to the student for whom it was meant for. She retrieved her hand back and answered the teacher who was glaring lasers at her through his glasses.

"Yes Mr. Anderson??". "First, please stand up while you speak to a teacher" Mr. Anderson pushed his glasses up. The student sighed and stood up, putting away the paper she had in her pocket.

"Would you please answer the question which is on the board??". Ms. Cujoh read the question. She then rolled her eyes and read the question the teacher had written in white.

Prove that Cos2x = Cos^2x - Sin^2x

"What the??" The teenager reacted.

Mr. Anderson's gaze never left Ms. Cujoh. He was observing every move of her of how she was looking around the classroom, silently asking for help and her friends were giving her a shrug or a thumbs-down.

"Ms. Cujoh. We don't have all day"

After that, she knew that she would have to answer the question by herself.

"I-It's simple Mr. Anderson. Multiply both the sides by zero, and there you go. LHS = RHS".

The students roared with laughter at the teenager's statement. Mr. Anderson was actually expecting such kind of an answer from the rowdy teenager.

"Silence everyone" the teacher stomped his ruler on the table and the room became quiet. But it didn't last for long as the bell finally rung, meaning that it was time for dispersal.

The students quickly packed their bags by shoving their books and stationery inside and exited the classroom, scrambling on their feet. The whole classroom became empty, leaving the astute teenager and the stoic teacher alone.

The silence in the room grew kind of tense. So the girl started to pack her bag and leave the class. Until.......

"Jolyne". The teenager looked at the teacher. She really wanted to leave the vacated classroom and go home. Also she had successfully survived today's class. Now what was the problem??

"Your behaviour is getting worse day by day. And I'm afraid that you might need some extra tutelage for your discipline and your academics yet again".

The teenager rolled her eyes once again. She then placed one of her hands on her hip and looked at the teacher with disinterest.

"Give me a break Mr. Anderson. My answer was correct". "You can argue with the ones who will correct your answer sheet Ms. Cujoh. I'm definitely not the right person to go against with your illogical statement".

Jolyne held her fist tightly. She really wants to punch her teacher square in the face and make sure to get the glass shards of his glasses cut his eyes. But she decided to bottle her anger for now.

"Your discipline is very concerning. Also you are not taking your studies seriously. Even the previous teachers whom I had appointed to improve your etiquettes returned home with a broken nose and teeth" he said to himself.

Jolyne smirked to herself with pride.

"So I have decided to change my plans" he spoke.

"Ha!! So you finally understand, Mr. Anderson" she then crosses her arms and sits on the table while putting her leg on the other. "You will never be able to transform me into nerds like you want to. I'm what I want to be and I don't give a fuck to anyone who wants me to change".

"We will see about that" he then pushes his glasses. "Huh??" Jolyne raises her eyebrow at him. What was this thick-skulled teacher really planning to do to her??

"If it's another teacher, then do keep some extra bandages for the injuries, because I'll not go easy on them" she then cracks her knuckles.

"Now that's where I stop you Ms. Cujoh".

Jolyne then glared at the teacher.

"I'm not going to appoint a teacher to straighten you up. This time, it's a student of your age. Your own classmate".

Jolyne was slightly taken aback. A student?? Of her age?? And the student of her own classroom?? But she barely knew anyone who was good at both discipline and studies. The only persons she was aware of were mostly the divas, the jocks and the bullies.

"W-Who is this student you are talking about??" Jolyne asked. "You will meet her outside the school premises. She is actually one of my best students and has been topping her class for the past 3 years" Jolyne witnessed a sense of pride in Mr. Anderson's tone.

Jolyne became a bit nervous. Who was this nerd whom she hadn't even notice that she was in her school and her class?? And also, a girl??!!

"You might be wondering--'why a girl??' It's because I really want you to get along with her and learn whatever she teaches, and her being the same gender as you would be a good boost".

Jolyne then sniggered and got up from the table. "Just because she is a girl and is of my age, doesn't mean that I can't afford a bruise or two on her".

"I had already warned her about your..........tendency and she is perfectly fine with it. So do whatever you want to do to her, she will be alright with it as she was the one who chose to take the risk and tutor you".

What a daring one for a nerd......Jolyne thought.

Mr. Anderson then looked at his wrist watch. He then spoke "It's getting late. You may leave now".

"Oh thank you, Mr. Anderson" she said in a very sarcastic manner and she even bowed to add to her mockery as she left the class.

"And another thing Ms. Cujoh". Jolyne then looked behind at her teacher lethargically. Can the teacher just leave her already before she actually punches him??

"If you hadn't been the daughter of Dr. Kujo, I would have already expelled you. But I'm still giving you a chance because I'm sure that you too can be as disciplinary as your father".

"Alright first of all, Dad was a delinquent when he was my age. And second of all, he would have already punched your face and shove your stupid glasses into your mouth just to make you shut up if he was at my place" and she then goes away without saying anything else or looking back.

Mr. Anderson sighed as he stacked the sheets and registers he had to take home and correct. After he was done, he looked at the empty classroom once more, especially at the seat where the spiteful teenager sat.

"Hopefully you know what you are doing, Noriko.........."

*One minute time skip, brought to you by Binod*

"Stupid Anderson, he just doesn't know how to mind his own fucking business!!!!" Jolyne angrily stomped out of the premises. She then turned around to look at the clock which was installed on the school building.

It's 5 already. I need to return home fast. Dad is coming home earlier than his usual time so I better reach home before him..........

"Umm....Jolyne Cujoh??".

"What?!!" She turned to the voice. Jolyne relaxed her eyes a bit when she saw who was the owner of the voice.

It was a girl wearing the same uniform as her. The only difference was that she was comparatively shorter than Jolyne and was fairer than her.

She had her red hair down which was a bit longer than shoulder length and purple eyes, one red wavy bang was framing her pretty face and her lips were a glossy, cherry red.

"Do I know you??" Jolyne raised her eyebrow at the unknown girl.

"Not yet, but you will. My name is Noriko. I'm the one who is supposed to mentor you" she then goes closer to her and offers her hand to shake. "It would be a pleasure to get to know you better".

Jolyne observed the girl's hand. She had her nails perfectly trimmed but there was no polish on them. Jolyne then looked at her face and asked.

"Japanese??" She asked. "Yeah" Noriko replied. Jolyne then smirks and shakes her hand firmly.

"Then we will get along pretty well" she said. Noriko then smiles and brushes her bang behind her ear with the other hand. "I'm glad you think so" she replied.

Both of the girls let go of their hands and then they walk home beside each other.

"By the way, why did you say that we will get along well, judging by my nationality??" Noriko asked the taller girl.

"You see, I'm a Japanese myself".

"You are??" Noriko asked, a bit surprised.

"Yeah. I know. I may not look like that but yes, Japanese is one of my nationality" Jolyne replied.

Noriko then snickered and asked "Just how many nationalities do you have??". "Well, my mother is from Florida and my father is Japanese, Italian and British" Jolyne calculated.

"Wow, so many citizenships" she commented. "Haha, it's not that cool as it sounds like. Trust me" Jolyne joked. Noriko chuckled and slapped the taller girl's shoulder lightly which made Jolyne chuckle.

"So Noriko??". "Yes??". "Why did you choose to mentor me?? You do know that I have a certain.........reputation at school".

"I am aware of everything you do in school. Mr. Anderson has updated me everything on you".

"*Ugh* That fucker. What is he, a spy?? Did he only find me to spy on?? And not the ones who actually harass girls or bully lowerclassmen behind the school??"

"Actually, I should be the one at whom you should be angry at". "Huh??" Jolyne looked at her.

"Wait, don't tell me--". "Yes. I was the one who told him to note everything you do".

"*Ugh* Noriko~!! I didn't expect this from you~" Jolyne whined. "Sorry, but this was the only way to plan on how to get you into discipline just like how Mr. Anderson want it". "Fuck that teacher!!!" She then kicks a stray rock on the road which goes very far.

"You seem to be very strong". "Heh, thanks. It's not much, but I can surely lift a heavy table pretty easily". "I'll take a note of that. It might come handy to me when I need to move a table" she chuckled. Jolyne then laughs and Noriko too joins her.

After their laughter died, Noriko spoke.

"You are really fun to be with". "Thanks. I had assumed that you were going to be just like Anderson but you are one lively person" Jolyne said.

"Oh really?? Thank you. Actually, no one has said that to me before" she then brushed another strand of her hair behind her ear.

"What do you mean??" Jolyne asked. "I don't really have friends. Only a few classmates talk to me but only when they need my assistance".

"Huh?? How is that even possible?? You are such an amazing person. Also, you are very pretty. You should have at least a boyfriend with you".

"I don't have a boyfriend". "Oh......a girlfriend perhaps??". "No.....". "Oh, you love a trans??". "What?? No. I'm not interested in anyone for now". "Ahhh....I see".

"Say, do you have a boyfriend??". "My status is also as same as you. Even I'm not really interested in getting into relationships. It will just..........get in my way, that's why". "I understand. Love is.......a very new subject for me" Noriko said.

"Yeah.....me too.....Hell, I don't even know why Dad fell for Mom in the first place just to leave her after that" Jolyne stated. "Huh?? What do you mean??" Noriko looked at Jolyne.

"My parents got a divorce after I was born. And according to the laws, both of my parents have the rights on me. So I visit one of them after every alternate month. This month I'm staying with my father" Jolyne explained.

"I see. Your father is Dr. Jotaro Kujo right??". "Yes, the famous marine biologist. The one who wrote thesis on starfish and other marine creatures. How interesting" Jolyne said, adding a dramatic sarcasm to it.

"Honestly, I'm a big fan of him" Noriko added, a light blush on her cheeks. "What?? Seriously??" Jolyne gaped.

"Yes. I have read every report and thesis he has ever written. The way how he observes and analyses the creatures is just wonderful. I bet that he is going to be the one who will discover a new species of marine life which humanity has never seen before" she spoke with amazement.

"Ha!!! You are flattering him now. My Dad is nothing like how you think of him" Jolyne argued. "Oh really??" Noriko raised her eyebrow.

"Yes. I know he is smart and all but he is not Einstein-level" Jolyne answered. "I never compared him with Einstein, Jolyne. It's just that I really think of him very fondly, and I admire him a lot" Noriko said.

"Yeah yeah, but don't fall in love with him. He might break your heart" she joked. Noriko blushed pink. It's good that Jolyne didn't notice it, otherwise it would have been very awkward.

"Well, here we are. My Dad's house" Jolyne stopped and showed Noriko her house.

"Wow, it looks very big" Noriko looked all over the lavish bungalow.

"Heh, these are the perks of being a Joestar" she said. "Joestar?? What do you mean??" Noriko asked. "You first come inside and then I'll tell you" and so Jolyne and Noriko entered the luxurious house. Both of them removed their shoes and Noriko looked around the house with an agape mouth and wide eyes.

"Wow......this looks so much better from the inside. Just how rich are you people??" Noriko commented and asked Jolyne.

"Well, there was a guy named Robert Speedwagon who was friends with my great-great-grandfather. He then became rich after finding many oil ores out of nowhere. After he died, all of his luxuries went to my great-grandfather and that's how all of our expenses are taken care of" Jolyne explained.

"That's amazing. You guys are literally royals" Noriko commented. "Yeah. And also my great-great-grandfather was a noble in England, so he was rich even before the Speedwagon guy came" she added.

"I see" Noriko then sat on one of the loveseats. "Being a Joestar must be very lucky right??" Noriko asked. "If you think like that, you are gravely mistaken" Jolyne told her.

"What?? Really?? I can't believe this. But you guys have everything!!!.............I guess everything has it's pros and cons" Noriko told herself.

Jolyne was about to speak more, but then the bell of the house rang.

"That must be Dad. You stay here Noriko. I'll be right back" and then she goes to check the door. Since Noriko had nothing to do for now, she looks at her skirt and straightens it up a bit.

Jolyne on the other hand opens the door and sees her father standing outside, just as she expected.

"Hello Dad" Jolyne greeted. "Hello Jolyne, how was school??" Dr. Kujo entered the house and Jolyne stepped away to give him some space.

"Meh, it was fine. Mr. Anderson barked at me as usual" Jolyne told him. Mr. Kujo sniggered a bit.

"He doesn't seem to stop critisizing you, does he??". "Of course not. I bet he won't even shut up even after he is an old man and cannot even lift his hand to push his oversized glasses up" she mocked.

Dr. Kujo smiled a fraction as he removed his shoes and his coat. While he was putting his shoes on the rack, he noticed another pair of school shoes which were similar to that of Jolyne's.

"Jolyne, have you brought a classmate here?" Dr. Kujo asked.

"Uhh yeah. She is actually my mentor who is supposed to tutor me, starting from today. I hope you don't mind if she stays here".

"No, not at all" The professor replied. He then hung his coat up and headed to the living room, her daughter following her.

As soon as Mr. Kujo entered the room, he spotted the red-haired girl. She looked awfully familiar to him which made the professor stop on his tracks.

"Dad, what's wrong??" Jolyne spoke from behind her father.

"Noriaki........" He whispered.

"Noriaki?? No Dad, her name is Noriko"

Jolyne corrected.

Even her name is similar to his...........

Mr. Kujo continued to stare at the red head who was scribbling something in the notepad on her lap with a pencil.

That red hair, those amethyst eyes, and how she was concentrating on her work, everything Mr. Kujo had seen already a long time before. It all felt like Deja Vu to him. And he clearly remembers the person whom this girl reminds him of. He was his highschool lover after all.

"Noriko-chan" Jolyne calls her. Noriko looks up from her notepad to her friend.

"Here is my father, who is also your 'sole idol'. And Dad, this is Noriko, my friend" Jolyne introduced them to one another.

(Part 2)

#jojo's bizarre adventure #jotakak #jotaro x kakyoin #noriaki kakyoin #jotaro kujo #jolyne cujoh #jjba genderbend #female kakyoin #Spotify #what I wrote

25 notes · View notes

ia-bi-tia · 4 years

Text

Kris’ Julian balkan turbofolk playlist

Ilija Devoraković

I included translations <3

1. https://www.youtube.com/watch?v=hqf5bniwyFc

VESNA ZMIJANAC -PUNISH ME, PUNISH

I was taken away by a scoundrel in my craziest years forgive me, my darling I come back to you, both sinful and alone

Ref. 2x Punish me, punish me if my eyes lie let them, let them prove it to you my lips and my body when my breasts are your pillow

I'm tired of life my face is so pale understand me this time I am your sinful child

Ref. 2x

2. https://www.youtube.com/watch?v=ETOut1QsJUs

CARA MIMI - MAGIC

I feel love in my body and I know I would surrender to you I'm waiting for you and me to get drunk to understand that it is not a manipulation

Heat on the body, take me whole you be a guy, be a lover at work you will seek better, you will lose because you chose the best

And how it seems to you when I throw spells and when I dress in a sexy bikini you are magically attractive, it seems to me but I'm not one of those who would say sorry

I feel love in my body and I would never refuse people like you I'm waiting for you and me to relax to realize that I haven't drunk too much yet

And how it seems to you when I throw spells and when I dress in a sexy bikini you are magically attractive, it seems to me but I'm not one of those who would say sorry

I feel, I feel, so it seems to me if you weren't the one to say sorry but I like to hear, I like to be poisoned those tiny crumbs of love

And so say you're the one, say you're the only one because you were created for me, tailored for me just to my liking, and say it once that you will be mine

And how it seems to you when I throw spells and when I dress in a sexy bikini you are magically attractive, it seems to me but I'm not one of those who would say sorry

3. https://www.youtube.com/watch?v=_q2tG6ieH7U

BANE BOJANIĆ - ONLY DRUNK I CAN GET OVER

Without her, everything would fit in one glass And with her, two lives are not enough Play me only sad songs Because she was everything to me

Because of her, I would throw pride in the mud It also turns stone into gold I can't imagine life without her Because she was everything to me

Only drunk can I get over This night is long and fate is evil I steal days from God, and I still love her Each one I love reminds me of her...

4. https://www.youtube.com/watch?v=qC-ovL31Si8

MITAR MIRIĆ - DOBERMAN

Ref. When you need someone to protect you I'll be a Doberman I will be your shadow and on fidelity trained

Over me alive others will not get you so doberman guards the territory

Worth the risk of losing everything in two words he would describe you worth the risk, something best i'm crazy enough just looking for trouble

In the city, in the club wherever you show up you do nothing and make chaos you have dangerous guys on target and you have your hero to save you

Ref. 2x

There is no red bull either like kisses are mine you will fly without wings when I conquer you

Worth the risk of breaking to ruin everything in my life it is clear as day, you are special there is no joke with me everyone should know that

Ref. 2x

5. https://www.youtube.com/watch?v=q6YowZdfpEc

SEKA ALEKSIĆ - BLACK AND GOLD

You don't have to be pretty to be right for me you don't have to be mine I would steal you anyway

An artist in love if you are then let me know to paint on me landscape with bare hands

Ref. 2x Mix black and gold tonight paint on my skin like on a canvas touch me with your lips on my body and neck so then put me in the frame to know that I belong to you

You don't have to be strong you don't need a mountain of money you don't have to be nice to take you home

It's enough that you're here when I need a little dice paint me again even if it was a copy

Ref. 2x

......

anyway, thanks to @pieniharmaakani for encouraging me <3

#julian devorak #arcana shitpost #y'all i have a whole alt universe where the arcana is set in the balkans

14 notes · View notes

wisdomrays · 4 years

Text

TAFAKKUR: Part 242

MUSLIM CONTRIBUTIONS TO MATHEMATICS: Part 2

Abu Kamil (about 850–930), an Egyptian mathematician, wrote the Book on Algebra which consists of three parts:

(1) Solutions of quadratic equations,

(2) Application of algebra to geometry,

(3) Diophantine equations.

He improved the work of Khwarizmi and applied algebraic methods to geometry. His research was on quadratic equations, multiplication and division of algebraic quantities. His work also includes addition and subtraction of radicals. He found the following formulas:

ax.bx=abx2; a(bx)=(ab)x; (10–x)(10–x)=100+x2–20x

Abu Kamil also wrote the Book On Surveying and Geometry, which was intended for government land surveyors. There, he stated the nontrivial rules for calculating areas, volumes, perimeters, and diagonals of different objects in geometry

Ibrahim ibn Sinan (908–946), a grandson of Thabit bin Qurra, was both an astronomer and a mathematician. Fuat Sezgin writes, "He was one of the most important mathematicians in the medieval Islamic world." He studied geometry, and his work on calculation of the area under the graph of a parabola is especially appreciated. Going further than Archimedes, he introduced a more general method of integration.

Abu Bakr ibn Muhammad ibn al-Husayn al-Karaji (953–1029), also known as al-Karkhi, is regarded as the first person to have developed algebraic operations without using geometry. One of his major works was Al-Fakhri fi'l-jabr wa'l-muqabala (Glorious on algebra). Historian Woepcke recognizes Al-Fakhri as the beginning of the theory of algebraic calculus. Here, al-Karkhi introduced the monomials x, x2, x3, ... and 1/x, 1/x2, 1/x3, ... and explained product rules among them. Moreover, he was the first to find the solutions of the equations ax2n+bxn=c. Al-Karkhi proved the sum formula for integral cubes by using the method of proof by induction, and hence became the first to use this method.

Abu'l Hasan ibn Ali al-Qalasadi (1412–1486) was an Andalusian Muslim mathematician. His main contribution was to introduce algebraic symbolism, and he used short Arabic words for his symbols. For example, he used the symbol for the sound "sh" from the Arabic word meaning "thing" to represent what we call x, the unknown.

TRIGONOMETRY

Khwarizmi also contributed to trigonometry. He established accurate trigonometric tables for sine and cosine, and he was the first to introduce tangent tables. [16] In 1126, these works were translated into Latin by Adelard of Bath.

Al-Battani or Albetagnius (about 850–929) was a Muslim astronomer and mathematician. In his research on astronomy he used trigonometric methods which were a lot more advanced than the geometric methods used by Ptolemy. He introduced trigonometric ratios. For example, for a right triangle with adjacent sides a and b, he gives the formula b sin(A) = a sin(900 – A), which is equivalent to tan A = a/b. He was the first to introduce the cotangent function.

Muhammad Abu'l Wafa (940–998), born at Buzjan in Khorasan, introduced the use of secant, cosecant and tangent functions. He gave a new method of constructing sine tables. He calculated sin(30^0) with an accuracy of up to eight decimal digits. He improved spherical trigonometry and proved the law of sines for general spherical triangles. In particular, he developed the half/double angle formulas:

2 sin2 (x/2)=1–cos x; sin 2x=2sin x cos x

He was the first to introduce the notion of secant and cosecant, and hence completed the list of all six trigonometric functions.

Abu Abd Allah Muhammad ibn Muadh Al-Jayyani (989–1079) was an Arab mathematician from Andalus. He was the author of The Book of Unknown Arcs of a Sphere which was "the first treatise on spherical trigonometry." Here he mentioned formulas for right handed triangles and law of sines. He also stated the formula for the solution of a spherical triangle in terms of the polar triangle. He had a strong influence on the West.

Another outstanding mathematician Nasir al-Din al-Tusi (1201–1274) wrote Treatise On The Quadrilateral, considered the best book on trigonometry written in medieval times, later translated into French by Alexandre Carathéodory Pasha in 1891. In his book al-Tusi made enormous advances in plane and spherical trigonometry. The Dictionary of Scientific Biography states, "This work is really the first in history on trigonometry as an independent branch of pure mathematics and the first in which all six cases for a right-angled spherical triangle are set forth." The well-known sine law is also stated in this work: a/sin A = b/sin B = c/sin C.

Ghiyath al-Din al-Kashi (1393–1449) produces sine tables of up to eight decimal places. In 1424, he computed 2π to an accuracy of sixteen decimal digits. He wrote a very impressive book on mathematics: Miftah al-Hussab (Key to Arithmetic). His main purpose in this book is to provide sufficient knowledge of mathematics for those who are working on astronomy,surveying, architecture, accounting and trading. He also describes how to find the fifth root of any number.

Unfortunately, the contributions of Muslims often go unrecognized. Muslim scholars contributed to science in many aspects such as mathematics, astronomy, geography, philosophy, medicine, art, architecture and so on. However, today few realize that in that era Islam played an important role in all aspects of life. Europe faced losing the works of major scholars, but as a result of their translations into Arabic most of this scholarship not only survived, but was further developed. Inspired by the Qur'an and hadiths, Muslims sought knowledge for the benefit of humankind. As the Qur'an says, "Are those who know equal to those who know not?"(Zumar 39:9). We should appreciate the scholars of all eras for their contributions to science.

1 note · View note

madewithonerib · 4 years

Video

youtube

Voddie Baucham on the "Sissified Needy Jesus?" [Revelations 19]

To the critics of this video.

No doubt, Voddie fully understands that GOD's love was also the biggest part of HIS SON’s crucifixion.

However, he certainty refuted those false Gospel claims that JESUS is a sissy, & that GOD is only about love.

Pastor Voddie Baucham is a solid bold preacher that reveres the LORD & HIS WORD.

Most certainly, there will be many in the world [and even some in the Church,] that will throw up smoke screens & ad hominem attacks, calling pastor Voddie "legalistic, judgmental pharasitical, critical, and/or negative."

Fact is the pastor Voddie is just being godly, & submitting to GOD's WORD.

Don't ever let anyone discourage you from valuing GOD's WORD, & desiring to be obedient to HIS WORD.

Music 'Treachery' by 'Trevor Jones'

Sermon Transcripts

Brokenness to our sin is appropriate because our sin is an affront to a Holy GOD.

Listen to what David says,

"Against YOU & only YOU I have sinned; & done what is evil in YOUR sight, so that YOU may be justified in YOUR WORDS & blameless in YOUR judgments.

Behold I was brought forth in iniquity & in sin did my mother conceive me.

Behold YOU delight in truth & in the inward being. YOU! YOU teach me wisdom in the secret heart.

In other words, here's what GOD desires & GOD delights in. And yet, here I've sinned.

Notice he didn't say against Bathseba I have sinned, or against Uriah I have sinned.

No against YOU GOD!!

And ultimately against YOU only I have sinned.

Because who was Uriah but the man that YOU created.

Who was Bathseba? A servant of the most high GOD!

I sinned against YOU. A holy & a righteous GOD.

And that's what worries me about the Shaq (Shaquille O'Neal) & that worries me about the Rabels(?) of the world.

That's what worries me.

That's what worries me about those who don't want to preach on sin because people already know that they're bad...

No we don't (2x), we watch the nightly news & go those people are bad, not us.

We don't recognize that we have sinned against a Holy & Righteous GOD.

We don't get that, we don't see that.

Turn with me if you will, to Revelation 19, I just love this. Maybe it says something about my character, but I despise the picture that is painted in our culture of this sissified needy JESUS, amen?

And that's who HE is, a sissified needy JESUS (falsified version of JESUS in pop-culture).

HE's just yearning for you. HE's longing for you, HE wants friendship & relationship with you, HE needs you. Oh you're breaking HIS heart.

NO! HE's going to break you.

Newsflash, by definition,

GOD is self-sustaining, self-existent, & self-sufficient; therefore by definition: HE needs nothing.

GOD does not need you, & HE's going to prove it one day, because you're going to die & the world is going to keep on spinning at the same rate it was before you were here.

And somebody is going to get all your stuff.

Yeah HE's waiting for you alright. [3:33]

Revelation 19:11-20 | Then I saw heaven standing open, & there before me was a white horse. And its rider is called Faithful & True.

With righteousness HE judges & wages war.

HE has eyes like blazing fire, & many royal crowns on HIS head. HE has a name written on HIM that only HE HIMSELF knows. HE is dressed in a robe dipped in blood, & HIS name is The WORD of GOD. The armies of heaven, dressed in fine linen, white & pure, follow HIM on white horses.

And from HIS mouth proceeds a sharp sword with which to strike down the nations, & HE will rule them with an iron scepter.

HE treads the winepress of the fury of the wrath of GOD, the Almighty.

And HE has a name written on HIS robe & on HIS thigh: KING OF KINGS AND LORD OF LORDS. Then I saw an angel standing in the sun, & he cried out in a loud voice to all the birds flying overhead, “Come, gather together for the great supper of GOD, so that you may eat the flesh of kings & commanders & mighty men, of horses & riders, of all men slave & free, small & great.” Then I saw the beast & the kings of the earth with their armies assembled to wage war against the One seated on the horse, & against HIS army. But the beast was captured along with the false prophet, who on its behalf had performed signs deceiving those who had the mark of the beast & worshiped its image. Both the beast & the false prophet were thrown alive into the fiery lake of burning sulfur.

What is wrong with the world?! You.

I mean me too, but mainly you. (lol)

Continued: youtu.be/Ymh5PdvS-S0

#Revelations 19 #Voddie Baucham

3 notes · View notes

marshmellowbobcat · 5 years

Text

Since Rotten Tomatoes has decided to hide my review (similar to how KB and RT plan on hiding from fans) here it is in all of its unedited, written at 1am glory:

When did Veronica become an asshole?Okay, sure she can be thoughtless and manipulative at times, but an outright asshole? Somehow she’s regressed… except worse because she was better than this in high school. Now, she barely pays attention to her friends, mocks her boyfriend’s journey towards mental health, and seems to have lost the innate softness that fascinates when it peeks through.

Veronica herself says she doesn’t have a “keen sense of mercy,” however, the show’s previous strength was in highlighting the occasions where she does display it, the moments she chooses to bend. Now we’re to believe the softness is gone, replaced by a jaded individual in desperate need of the therapy she so often mocks.

And maybe that would have been interesting, if they bothered to explore how she got there. There’s no bridge between the righteous woman at the end of The Movie and S4 Veronica, whose “fight” now seems like hubris.

So fine, start us off with a “darker” Veronica. Show us her struggle against being the “defective gene” in her relationships. Except… One of the biggest sins of this season is that any emotional development on her part seems abrupt and is told to us, and not shown.

Instead of exploring her personal character, choices, and the emotional depth of her long term relationship, the “marshmallow of it all” if you will, those things are given little screen time in favor of what, quite frankly, is a lackluster mystery with 5,000 red herrings.

If you don’t figure out whodunnit by episode 1, there’s an error in episode 2 that gives it away. Hint: the guy you think dunnit, dunnit. Gone are the days when you felt this clever woman was smarter than everyone around her. There’s a woeful lack of outsmarting and “I know what happened” moments.

And you would think that not expending too much thought power in the mystery would give the show an opportunity to focus on Veronica’s relationship dynamics with Neptune’s complex and compelling cast of characters. Well, you would be wrong. Veronica’s personal “progression” is rushed, crammed in between long bouts of barking up the wrong tree. It’s incredibly frustrating.

She turns down a marriage proposal, spends a few episodes being self-righteous, borderline cheats on her boyfriend, and has an epiphany (accepting the proposal) all in rapid succession. The time that could have been used to examine her feelings (she could, for example, have spoken to her married, “family man” best friend Wallace about the proposal) is spent proving how “edgy” and “adult” this iteration of Veronica Mars is. It feels choppy, disjointed, and cheapens her journey.

(Side note: watching a 35 year old woman drop E on a random Tuesday with a stranger, a guy who has a history of attempted date rape, and her boyfriend who has been in rehab 2x for substance abuse isn’t “edgy,” it’s terrifying.)

And STILL, that would be fine. I probably would have clamored for a season five in hopes that it would be less problematic… if they didn’t torpedo what little progression she did make with the most lazy, unnecessary end to a favorite character. An end that felt like an afterthought. An end that was unworthy of 15 years and fan fueled franchise resurgence.

SPOILER:

I guess“Shock Value” is the watchword of 2019, and begs the question … is it shocking if no one is shocked? It’s getting old.

Veronica becomes a widow within hours of her wedding ceremony, effectively resetting her back to Season 1 Veronica. A woman who needs a catalyst for strength. A woman who needs trauma (not, you know, mental health care) to make her a force to be reckoned with.

The show glorifies her pain, it tells victims that nothing is within their control, and it underscores it with a (bullshit) voicemail from her deceased husband waxing poetic about how “strong” she is. As if to tell the audience: she’s over it, she rose above it, you should be too. There’s not even a mourning process for our protagonist, a year time jump skips over all those messy feelings she may have had.

This, too, feels lazy and rushed. It’s a disservice to everyone involved, but especially the viewers who tuned in for a cool mystery and interesting character dynamics and instead got an uninspired whodunnit and a heroine that is often hard to root for.

TLDR; #FormerMarshmallow

#veronica mars spoilers #veroncia mars season 4 spoilers #veronica mars #rotten tomatoes review

112 notes · View notes

sanguinarysanguinity · 5 years

Text

Navigatio Britannica: Planar Trigonometry

Chapter 4: Of Trigonometry, Sections I - V

The first half of this chapter is Planar Trigonometry, which I learned in high school and have used on-and-off ever since; the second half of this chapter is Spherical Trigonometry, which I know nothing about. Consequently, I’m dividing this chapter into two parts -- before we let John Barrow attempt to teach me spherical trig (wish me luck!), I want to do a fast recap of what he has to say about planar trig...

Section I: Definitions

Everything is defined geometrically, on the unit circle, via a diagram that I have yet to find in this scan. (Also, even if I do find the page, I don’t have much hope that it will have been scanned correctly, since Google’s scanning machine can turn pages but not unfold them.) Happily, Barrow is pretty good about describing his figures in enough detail that I can reconstruct them as I go, which is the only reason I was able to understand anything in this chapter.

Not many surprises here, although I did learn that cosine, cotangent, and cosecant are the sine, tangent, and secant of the complementary angle, and likewise that the tangent of an angle is called the tangent because its physical instantiation lives on a line tangent to the circle. Also, Barrow defines the Verse-Sine, which was new to me: geometrically, it is the part of the radius that isn’t the cosine. (Algebraically, it is 1 - cosine.) Wikipedia says the versine was important to navigation, so I assume it will come up later.

Section II: Geometrical Constructions of the Tables of Sines, Tangents, Secants, &c.

In which we are instructed to build ourselves a unit circle, mark it off in 1-degree intervals, and construct ourselves a... well, it’s gonna look a bit like a ruler, but it’s going to measure 1 to 90 degrees, on several parallel scales: chords, sines, versed-sines, tangents, etc. To make this thing, you use your compass and measure the length of a chord for a 10-degree angle, then mark it on your chords-scale, and label it “10 deg.” Repeat for the other 89 degrees, and ta-dah, you have a chords-scale! Then do it again for sines, versed-sines, tangents, and so on. When we get to actually solving trig problems, how to use this scale is one of the three standard methods that Barrow is going to teach us.

Section III: Arithmetical Constructions of the Tables of Sines, &c.

First off, Barrow reassures us this is going to be easy-peasy, no need to panic -- which is our first cue that panicking will be required before we’re done.

But true to his word, Barrow starts out easy, using similar triangles to prove all the basic trigonometric identities: tan = sin/cos, sec = 1/cos, etc. All well and good, except it’s all done in proportions and nothing is called out by name, only by referring to various line-segments in his nowhere-to-be-found unit circle diagram. All of which makes it difficult to absorb at a glance, but once you finish decoding everything this is basically just SOHCAHTOA.

Then he proves two variants on the standard trigonometric sum/difference formulas (although he expresses them as proportions and via verbal descriptions, talking about the means of equi-different angles and the differences between them):

cos x = (1/2) (sin y + sin (y + 2x)) / sin (y + x)

sin x = (1/2) (sin (y + 2x) - sin y) / cos (y + x)

You can verify those via the standard trigonometric sum/difference identities if you want. (I did.) But they’re also pretty straightforward geometrically, if you take the time to very carefully reconstruct what his diagram must have been: in the end, it’s all just similar triangles. He then proves several corollaries -- which in hindsight are simple enough (just straightforward algebraic manipulations, multiplying everything by two, or both sides by the denominator), but sadly, I lost MANY HOURS to a rash of typos in them.

Then.

Oh, then.

All hell breaks loose as he endeavours to prove that a semi-circle has an arc-length of pi. I admit to not following this bit: I haven’t seen Newton’s notation for calculus since I was seventeen, when that one weirdo physics professor used it in lectures, and I didn’t really feel like re-teaching it to myself for this. Nor did I really want to get into re-teaching myself binomial expansions. Also, the type-face on all the fractions in the expansions was super-squinchy to read, and you know what, fuck it, I think it’s well-established that a semi-circle has an arc-length of pi, let’s move on.

The point of establishing that a semi-circle has an arc-length of pi is so that we can calculate the arc-length of one minute (simply divide pi by 10,800 minutes, easy-peasy), which we will then use as an approximation of the sine of one minute. ... Which, okay, I suppose if your angle is small enough and your applications are practical enough you can get away with that? But it makes the mathematician in me cry, I’m just saying. (Even as I admit that you really can get away with it for most purposes: according to my handy-dandy TI-84 Plus, pi/10800 differs from sin(1′) in the ninth significant digit. But Dr. Roberts and Dr. Chrestenson would never have let me get away with that shit, never mind that I also have an engineering degree and thus should be okay with this kind of ruthless practicality. In my soul there is a mathematician and an engineer battling to the death over questions like these, you simply don’t know how much shit like this wounds me.)

Anyway, once I finally got over my fit of vapours...

Now that we have an approximation for the sine of one minute, we can calculate the cosine of one minute via the pythagorean trigonometric identity, and then...

And now I want to cry again, because now we get to build our table of sines (and along with it, our table of cosines), minute by freaking goddamn minute, by using the above equations like so:

2 cos (1′) sin (1′) - sin (0′) = sin (2′)

2 cos (2′) sin (2′) - sin (1′) = sin (3′)

2 cos (3′) sin (3′) - sin (2′) = sin (4′)

...

Continue until I cry blood and the seas boil dry.

(At one point Barrow admits that it’s possible to build this table in 5-minute increments and interpolate the intervening minutes when you need them. While this reduces the task to 1/5th of the original, I still want to hug and rock myself and cry.)

Happily, I don’t need to cry, because Barrow includes these tables in the book? But someone cried blood to make those tables, and John Barrow wants us all to know it.

Section IV: Actual Trigonometry Problems, At Long Last!

A ton of sample problems, all worked three ways:

Geometrically: Basically, use a compass and straight edge and your scale-thingie of chords/sines/secants that you made earlier, and draw a triangle of the correct proportions. Then just read/measure your answer right off the actual triangle in question, ta-dah! No abstract math required, just pretty pictures!

Arithmetically: What you learned in high-school, using the tables that someone cried blood over but without calculators (although you can use logs if you want to skip ahead to chapter five for them!) God, it looks miserable and grindingly awful, and I admit I don’t have the strength of character to follow any of these calculations through to the end.

Magic, I mean, Gunter’s Scale: The instructions here are amazingly low-key -- use your chart-dividers (what tumblr calls a pointy-leg-man what it likes to make walk on tippy-toes across charts) to measure off an interval on one scale, and then drop that same interval across the appropriate second scale, and voila! You have found your answer!

Of course I wanted to know what this magical tool is!

Apparently it was a slide rule without the slidey parts -- you used your dividers to accomplish what the slidey bit does on a slide rule -- but with some extra scales especially chosen for the convenience of navigators.

Apparently these things were so common among navigators that they were simply called Gunters, and I WANT ONE SO BAD. Here’s a nifty article about them, complete with pictures, and did I say? I WANT ONE SO BAD. I collect old-school mathematical tools and I WANT ONE SO BAD.

Ahem.

Anyway, Section V is more trig homework, except now we’re no longer dealing with right triangles. I admit it, I skimmed this like fuck.

And ta-dah! That’s Planar Trigonomometry, according to John Barrow in Navigatio Britannica, or, A Complete Guide to Navigation, pub. 1750!

Up next: Section VI - Spherical Trigonometry, what makes William Bush cry. Do I have the fortitude to teach myself spherical trig? PLACE YOUR BETS NOW.

#navigatio britannica #john barrow #age of sail #trigonometry #long post

2 notes · View notes

reedreadsgreek · 3 years

Text

Romans 7:9–12

9 ἐγὼ δὲ ἔζων χωρὶς νόμου ποτέ, ἐλθούσης δὲ τῆς ἐντολῆς ἡ ἁμαρτία ἀνέζησεν, 10 ἐγὼ δὲ ἀπέθανον καὶ εὑρέθη μοι ἡ ἐντολὴ ἡ εἰς ζωήν, αὕτη εἰς θάνατον· 11 ἡ γὰρ ἁμαρτία ἀφορμὴν λαβοῦσα διὰ τῆς ἐντολῆς ἐξηπάτησέν με καὶ διʼ αὐτῆς ἀπέκτεινεν. 12 ὥστε ὁ μὲν νόμος ἅγιος καὶ ἡ ἐντολὴ ἁγία καὶ δικαία καὶ ἀγαθή.

My translation:

9 Now I was living without the law at one time, but when the command came sin lived again, 10 and I died and the command unto life, this command was found in me unto death; 11 for sin having taken an opportunity through the command, deceived me and, through it, killed me. 12 For this reason the law indeed is holy and the command is holy and just and good.

Notes:

7:9

δὲ is probably transitional (“Now”) and is omitted in most translations.

ἐγὼ is the emphatic subject of the imperfect ἔζων (from ζάω; most translations, “I was alive”). For χωρὶς νόμου, see note on verse 8. The temporal particle ποτέ (“formerly”; most translations: “once”) modifies ζάω adverbially. This time “without law” could refer to Paul’s childhood (ICC) or generally to mankind before the introduction of the law of Moses (Cranfield); it could also be both, the former serving as a metaphor for the latter. NIGTC, following Calvin, posits the period to refer to Paul’s time as a Pharisee prior to encountering Christ.

The feminine aorist participle ἐλθούσης (from ἔρχομαι) with τῆς ἐντολῆς forms a genitive absolute construction indicating antecedent time with the main verb ἀνέζησεν below.

αναζάω (2x), from ἀνά + ζάω, is literally, “I live again”; BDAG: “spring into life” (NIV, HCSB); NRSV: “revived”; NASB: “came to life”; NET: “became alive”. The verb probably indicates a prior state not of death but of dormancy. ἡ ἁμαρτία is the subject of the aorist ἀνέζησεν.

7:10

ἐγὼ is the emphatic subject of the aorist ἀπέθανον (from ἀποθνή̣σκω).

The subject of the aorist passive εὑρέθη (from εὑρίσκω; BDAG: “proved to be”) is ἡ ἐντολὴ. μοι modifies the verb as a dative of disadvantage. The article ἡ places the prepositional phrase εἰς ζωήν, indicating purpose, into attributive position with ἡ ἐντολὴ (NIV, NET: “the very commandment that was intended to bring life”).

The feminine near-demonstrative pronoun αὕτη is resumptive (“this same command”). εἰς θάνατον indicates result.

7:11

For ἀφορμὴν λαβοῦσα, see note on verse 8.

As in verse 8, the instrumental prepositional phrase διὰ τῆς ἐντολῆς could be taken either with participle above (most translations) or indicative verb below. Cranfield says, ‘the fact that διʼ αὐτῆς must be taken with ἀπέκτεινεν makes it natural to connect διὰ τῆς ἐντολῆς with ἐξηπάτησεν rather than with λαβοῦσα’. However, if διʼ αὐτῆς were absent, διὰ τῆς ἐντολῆς could be read with both verbs; its inclusion seems to indicate that διὰ τῆς ἐντολῆς goes with the participle. Furthermore, it is harder to see the law as the means of deception than merely the opportunity for it.

ἐξαπατάω (6x) is an intensified form of ἀπατάω (3x) “I deceive” (ἀπατάω is used in the account of the fall, Gen. 3:13 LXX). ἡ ἁμαρτία is the subject of the both aorists ἐξηπάτησέν (with direct object με) and ἀπέκτεινεν (from ἀποκτείνω, a liquid aorist). The instrumental prepositional phrase διʼ αὐτῆς, referring to τῆς ἐντολῆς, modifies ἀποκτείνω.

7:12

ὥστε introducing an independent clause is, “for this reason, therefore, so” (BDAG). As a conclusion to the section, most translations render it, “So then”.

The particle μὲν lacks the corresponding δέ, but there is probably an implied contrast with sin (EGGNT).

νόμος is the subject of an implied ἐστιν, and ἅγιος is the predicate adjective.

ἡ ἐντολὴ is the subject of an implied ἐστιν, and ἁγία, δικαία, and ἀγαθή are predicate adjectives.

#Romans #Romans 7 #law #sin #death

0 notes

mathematicianadda · 4 years

Text

Geometric Similarity of Functions https://ift.tt/3jnkPC3

I am a 16 year old high school student and recently I have written a paper on a numerical approximation of distinct functions. I have shown my teachers this and they do not understand it. My questions: Is this a valid theorem to use to estimate functions with differently based functions? Has something similar already been created? Is it all useful/publishable? Any tips on how to improve? I will give an outline but you can find it here: https://www.overleaf.com/read/xjqhfgvrcrbj

Definitions

Geometric similarity refers to the dilation of a particular shape in all its dimensions. Proofs of geometric similarity are included in congruence proofs of triangles with AAA (Angle-Angle-Angle) proofs. Knowing the sizes of all sides of both triangles: $\triangle{ABC}$ and $\triangle{A'B'C'}$, to find the dilation factor and prove geometric similarity the following must be true: $\frac{\mid A' \mid}{\mid A \mid} =\frac{\mid B' \mid}{\mid B \mid}=\frac{\mid C' \mid}{\mid C \mid}$.

Interpreting functions as shapes on the Cartesian plane and using geometry, geometrically similar functions can be calculated. Analytically this would imply for a function $y=f(x)\; \{x_0\leq x \leq x_1\}$ a geometrically similar function would be of the form $ny=f(nx)\;\{\frac{x_0}{n}\leq x \leq \frac{x_1}{n}\}$ where $n\in {\rm I\!R}$. This is because the function is scaled by the same factor in the $x$ and $y$ direction thus would be geometrically similar.

$y_1=\sin(x)\;\{0\leq x \leq 2\pi\}$ and $y_2=\frac{1}{2}\sin(2x)\; \{0 \leq x\leq \pi\}$" />

However to compare two functions which are distinct, multiplying $x$ and $y$ by $n$ will not suffice for proving similarity. The formula to find the dilation factor can be used to prove similarity between two functions. By describing a function geometrically it has three superficial 'edges' which can be represented as sets. Two of the edges are the two axis $x$ and $y$. The length of the side '$y$' is the $\max \{ f(x) : x = 1 .. n \}-\min \{ f(x) : x = 1 .. n \}$ and the length of the side $x$ is $b_1$-$a_1$ where $b_1$ is the upper bound and $a_1$ is the lower bound. Finally the third side of the function will be the arc length over the interval $\{a_1\leq x\leq b_1\}$. Another characteristic for two shapes to be geometrically similar is the area is increased by the dilation factor squared.Thus from the formula for the dilation factor for two similar triangles the following theorem can be derived:

Theorem Let $y_1\;\{a_1\leq x \leq b_1\}$ and $y_2\;\{a_2\leq x \leq b_2\}$ be functions whose derivative exists in every point. If both functions are geometrically similar then the following system holds: \begin{equation} \frac{1}{\big(b_1-a_1\big)}\int_{a_1}^{b_1} \sqrt{1+\bigg( \frac{dy_1}{dx} \bigg) ^{2} } dx= \frac{ 1 }{ \big(b_2-a_2\big) } \int_{a_2}^{b_2} \sqrt{1+\bigg( \frac{dy_2}{dx} \bigg) ^{2} } dx \end{equation} \begin{equation} \frac{1}{\big(b_1-a_1\big)^2} \int_{a_1}^{b_1} y_1 dx= \frac{1}{\big(b_2-a_2\big)^2}\int_{a_2}^{b_2} y_2dx \end{equation}

Similarity Between Distinct Functions

When describing a function as distinct it denotes that the functions have different bases, i.e. sinusoidal and exponential. As mentioned above, for geometric similarity to exist of a function $y=f(x)$ the resultant function will become $ny=f(nx)$. However if comparing functions of different bases, equations (1) and (2) are necessary to find the bounds of similarity. For example, the problem:

Find the bounds $b$ and $a$ where $e^x\;\{0\leq x\leq 1\}$ is similar to $x^2 $.

To see examples go to the above link. Any help would be much appreciated and apologies if this is crude mathematics.

from Hot Weekly Questions - Mathematics Stack Exchange hwood87 from Blogger https://ift.tt/2FYTbOm

#Blogger

0 notes

fromanartistsheart2 · 1 year

Text

I had a hard time

Writing this rhyme

Deciding which stories

To share that aren't gory

2x brain surgery survivor

B4 the age of 2

Family was worried

About what I could do

3 more surgeries

For an ongoing health condition

That doesn't have a cure

Of that, Drs r sure

Raised by a single daddy

Mother nowhere to be found

We heard nothing for years

Not a single sound

My mom came along

When I was just 12

And she is the bomb

About it I wrote a song

Engaged at 19

To a man that was obscene

Going to strip bars

When I would ask where u r

I caught him cheating

In a private room

And that was the end of

The relationship that was doomed

From Carolina

Down n the south

And to go with it

I have a smart mouth

Met my next migraine

Thru their ex

It was insane

That should have been my sign

At age 21, married then pregnant

I was stuck with him

Didn't know where to go

But it was easy to leave, people told me so

First child born with health issues

That I didn't have enuff tissues

For all my tears

Throughout the years

I was wrong to make

My child behave

And to my husband

Was expected to b a slave

After 6 years

Baby 2 came along

I stayed home with the kids

I thought he was working, but I was so wrong

He lied to me again

And expected me to give in

To join in with his sin

And lie to people again

Begging for food

Which I thought was rude

Begging for rides

Or he would have my hide

Abuse and tyranny

The whole time

It even inspired

Lots of rhymes

Met my new husband

Right here on Facebook

Writing my rhymes

Of the terrible times

Baby #3

And oh woe was me

The oldest out of control

And so my temper rolled

Family interfered

And told my child to disobey

My child was on the losing end

Now matter which way

He listens to the people

Who claim to go to the steeple

But expect him to b a sheeple

And listen to the wrong people

I was suppose to let

His behavior slide

And let him do what he wants

Because to him people lied

So in 2015

When things were past obscene

I met a new guy

And to me he didn't lie

He also wrote rhymes

And I responded most of the time

He was there for me

And my kids 3

So we wrote forth and back

And he never gave me slack

Told me like it was

And never lied because

He is a real man

And he came up with a plan

To rescue me

And my babies, 3

Over time we grew

And by now each other knew

That we loved the other

And had to b with one another

So n 2017

My son made accusations (obscene)

So the cops said we had to leave

And no longer have to cleave

The cops never investigated

Never tried, and it's them I hated

He has big connections

But none that I can prove, just suspections

He got the kids back

And kept them n slack

And now their grades they're failing

Like everyone that I've been telling

Everyday I live

It is for my kids

To save them from him

And everything he's did

He's tried to get me to stop the fight

So he can do everything not right

So that I won't report

And to threats he won't have to resort

So f*cked up

He has them

From what

He's done to them

P*ssed off?!

I'm way beyond

And of the court system

I am not fond

They tell me that my kids

R f*cked up beyond repair

And so I just should stop

And not even care

I should just go on

Live my life and forget

But if I do

I'll have lots of regrets

I tell u all these things

Cause it's for my kids that I breathe

And all the things my ex has done

It just makes me seethe

Fill up with anger

To the point of danger

And go full beast mode

And just explode

But if I do that

He will get what he wants

To prove I'm mentally unstable

And get the kids? I won't

Protect my kids?

The system won't

And my ex?!

I KNOW he won't!

So my heart breaks

With everything he takes

Away from my kids

Just like to him, his family did

Heartwarming Hymns

0 notes