#Proofs of Trigonometric Identities

[tooies]

today tooie will verify proofs using trigonometric identities

[cergisoft]

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.

[firstbuzz]

US Teens Discover New Proof For The Pythagoras' Theorem, Stun Mathematicians

Two American high school students have stunned mathematicians after they claimed that they discovered a new way to prove Pythagoras' theorem by using trigonometry- a feat mathematicians thought was impossible, Guardianreported.

Calcea Johnson and Ne'Kiya Jackson, who are seniors at St. Mary's Academy in New Orleans, presented their findings on March 18 at the American Mathematical Society's (AMS) Spring Southeastern Sectional Meeting.'Their groundbreaking lecture from the research is historic. High School students are generally not presenters at the American Mathematical Society Meeting,'' the school's announcement notes.Notably, the 2,000-year-old Pythagorean theorem states that the sum of the squares of a right triangle's two shorter sides is the same as the square of the hypotenuse, the third side opposite the right angle. Students around the world learned the notation expressing the theory as a2+b2=c2.  However, mathematicians have struggled to find a definitive proof for the theorem which would not only show that it works but explain why it does.''In the 2000 years since trigonometry was discovered, it's always been assumed that any alleged proof of Pythagoras's Theorem based on trigonometry must be circular,'' they told an audience at the American Mathematical Society Southeastern Regional Conference.''In fact, in the book containing the largest known collection of proofs (The Pythagorean Proposition by Elisha Loomis) the author flatly states that ‘There are no trigonometric proofs because all the fundamental formulae of trigonometry are themselves based upon the truth of the Pythagorean Theorem.''The students further said they can prove the theorem by using trigonometry and without circular reasoning. But "that isn't quite true," the teenagers wrote in the abstract. "We present a new proof of Pythagoras's Theorem which is based on a fundamental result in trigonometry — the Law of Sines — and we show that the proof is independent of the Pythagorean trig identity sin2x+cos2x=1."However, the findings have not yet been accepted into a peer-reviewed journal. According to Live Science, it's still too soon to say whether their proof will ultimately hold up.

[devphilamaths]

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

[wodenscild]

Deriving the trigonometric identities

Leads to: Proving De Moivre's Theorem

Useful and neat solutions to large, seemingly clunky and messy expressions, can often be found with the use of either the trigonometric identities. Here, we will derive the angle sum and difference identities, followed by the double-angle identities, and the product identities. And with these, we will be able to prove De Moivre's Theorem through proof by induction in another post (it is sorta too late rn to do it jdshk)

Angle sum and difference identities

To start, the angle sum and difference formula. This is not too hard to find using matrix transformations. If we had a transformation, M1, which rotated the plane by A, and another, M2, by B, then applying one after the other would result in a rotation of A+B. i.e. M2 times M1 would give a rotation of A+B.

Here we have found the cosine and sine of A+B. But what about A-B? This one is a tad more fiddly to find out. Suppose we wanted to go backwards- clockwise around the plane- how would we do that? We need to define subtraction in terms of addition. What is subtraction at its core? The addition of with negative number. On the plane, a negative angle is defined as an angle between pi and 2pi. i.e. an angle which goes below the x-axis. As this is just a sign change, changing the height above the x-axis (i.e. changing position in regards to the y-axis), then we can get the following definition:

And so, we have enough information for us to find our double angle identities, and our product identities...

Double-angle Identities

Product Identities

(I have made a typo here, it should be [1]=[2] and [3]=[4], not [1]=[3] and [2]=[4] dsjfhkjsdl)

[bloggulf220]

Mr. Mac's Challengesmr. Mac's 6th Grade

Mr. Mac's Challengesmr. Mac's 6th Grade Language Arts

Mr. Mac's Challengesmr. Mac's 6th Grader

These seem like obvious things but just trying to help people with the learning curve.

This feature is not available right now. Please try again later.

Anker Tests Reading / Language Arts Skills. More Reading Comprehension More Spelling (10 words each) More Homophones More Analogies. More 6th General Math. By Mr Mac. Now contains Australian/British English and US English spellings. A great visual prompt for students to implement the 'Super Six' comprehension strategies when reading or viewing texts. great poster display for classrooms - text and images.

Elementary Math Skills

Mrs. Victoria Rose

This class will address 3-6th grade mathematical skills in preparation for pre-algebra.

Pre-Algebra

Mr. Mac Ogilvie 1

STARS offers a Pre-Algebra course using the Saxon Math 8/7 book as a resource. This course provides an excellent summary of the basic skills required to move into Algebra. It can be considered to be a Middle School Math course that bridges elementary school to high school math. Public schools seem to be moving some students into Algebra as early as sixth grade depending largely on the skill level of the student. But the movement into Algebra at any grade level requires the exposure to a wide range of general math knowledge and the mastery of key skills essential to the ability to be successful in Algebra. This course provides that knowledge and the ability to master those critical skills. The course includes coverage of basic geometry and probability and statistics. The STARS instructor at this time has an extensive background in teaching both middle school math and Algebra both in public schools and here at STARS.

Need: Saxon 8/7 Homeschool Kit- http://www.rainbowresource.com/product/Saxon+Math+8-7+3ED+Homeschool+KIT/024434

Mr. Mac's Challengesmr. Mac's 6th Grade Language Arts

Algebra I

Mr. Mac Ogilvie

This course will use the Saxon book, Algebra 1, to provide a comprehensive teaching of the fundamental aspects of problem solving. It offers a substantial review of pre algebra fundamentals while also offering coverage of area, volume, and perimeter of geometric figures. Major topics include evaluation of algebraic equations, thorough coverage of exponents, polynomials, solving and graphing linear equations, complex fractions, solving systems of equations, radicals, word problems, solving and graphing quadratic equations, solving systems of equations, and solving equations by factoring.

Textbook: Saxon Algebra 1 Homeschool Kit http://www.rainbowresource.com/product/sku/000628

Algebra II

Mr. Mac Ogilvie

This course will use the Saxon book, Algebra 2, to provide a comprehensive teaching of the fundamental aspects of problem solving. It offers a substantial review of all topics in Algebra 1 and then moves on to cover these topics at an advanced level. Major topics include the solving and graphing of linear and quadratic equations, factoring, a variety of types of word problems, solving quadratic equations by completing the square, solving simultaneous equations with fractions and decimals, complex roots of quadratic equations, solving systems of nonlinear equations, graphing and solving a system of inequalities, exponential equations, and review of key geometry, probability and statistics topics.

Textbook: Saxon Algebra 2: Homeschool Kit Third Edition

Jacob’s Geometry

Ms. Enjoli Stith

3rd Edition. This is an excellent geometry course with clear explanations of geometric concepts, including plenty of practice with proofs (informal and paragraph). The second chapter (six lessons) is devoted to logic in preparation for constructing proofs. Topics build incrementally and each practice set assumes knowledge gained in previous lessons in order to construct proofs. The author has set his text up to include three sets of problems with each lesson so as to present the basic concepts in Set I exercises, applications in Set II exercises, and extension of concepts in Set III exercises. Finally, there are Algebra reviews located at the end of most chapters in the student textbook. An appendix contains all presented theorems and postulates. After a thorough study of Euclidean geometry, a single chapter of four lessons presents non-Euclidean geometries. SAT math problems have also been included in exercise sets. The teacher’s guide contains lesson plans, black line masters, and answers to all exercises. The test bank contains two tests for each chapter, a mid-term, a final, and solutions.

Textbook: Geometry: Seeing, Doing, Understanding, 3rd Edition

Advanced Math (Pre-Calculus and Trigonometry)

Ms. Enjoli Stith

Advanced Mathematics fully integrates topics for algebra, geometry, trigonometry, discrete mathematics, and mathematical analysis, to include trigonometric equations and inequalities, the unit circle and trigonometric identities, conic sections, logarithms and exponents, probability and statistics, complex numbers, functions and graphs, sequences and series, and matrices. Graphing calculator applications are developed to facilitate calculations and enhance in-depth understanding of concepts. Word problems are developed throughout the problem sets and become progressively more elaborate. With this practice, high-school level students will be able to solve challenging problems such as rate problems and work problems involving abstract quantities. Conceptually oriented problems that help prepare students for college entrance exams (such as the ACT and SAT) are included in the problem sets. Students complete 2-3 lessons per week.

Mr. Mac's Challengesmr. Mac's 6th Grader

Note: This is a two year class. Part 1 covers lessons 1-66, Part 2 covers lessons 67-125.

[countrynerddancer]

What people think being good at math means...

- capable of reciting the multiplication tables in 144 seconds flat

- never makes addition or multiplication errors

- memorized all the trigonometric identities and laws and the graphs of every function

Versus

What it actually means:

- understanding the behavior of a function by thinking about it a bit

- memorizing methods and ways to know when to apply which one

- following proofs, proving things yourself

- understanding how to do matrix math and solve systems, isolate variables, and manipulate equations

- identifying information within a word problem/application and figuring out how to relate variables

Basically, what you were graded on and what made people "good" in middle or high school isn't really necessarily the most important thing come calculus, economics, statistics, or any science dealing with data.

And I've found that often, the people who say boys are naturally better at math tend to be thinking of the first kind, not the second.

[mathsodology]

Compositions of trigonometric and inverse trigonometric functions (35)

Compositions of trigonometric and inverse trigonometric functions (35)

Proof of (35). Method 1. We can take the reciprocal of (17): Method 2. Let where and , , i.e. What we want is in terms of , so our strategy is to express in terms of using trigonometric identities.

[maximuswolf]

Struggling with trig identities and proofs via /r/calculus

Struggling with trig identities and proofs

I’m learning about trigonometric identities and proofs now, and I don’t really get how to do it or really understand it at all. I’d like to be able to take calculus next year, and I know that the fundamentals are essential, so I don’t want to fall behind.

Are there any supplementary resources I can use to help reinforce trig identities and proofs? Preferably free and online.

Submitted February 25, 2021 at 11:27AM by xMultiGamerX via reddit https://ift.tt/3koX2UL

[hollohat]

Hello, tumblr!

Today, I’m going to talk about AP Calculus! It has a reputation as one of the hardest AP courses, not without desert; it is heavy with concepts and requires a high degree of proficiency in all the math that comes before it. It is also enormously useful for a variety of fields, from architecture to medicine, and can be a lot of fun to do! Some tips:

Prepare

Most AP Calc courses come after a substantial list of prerequisites: Algebra 1 and 2, Geometry, Trigonometry, and whatever your school calls the mish-mash of topics falling under pre-calculus. It is imperative to be comfortable with these when you start; calculus uses all of them.

Specifically:

Know. The. Trigonometric. Circle. Know it like the back of your hand (if you are someone who studies each detail of your hand carefully like the weirdo who came up with this saying)

Make sure you know the trigonometric identities too, back and forth.

You will need the formulas from Geometry. These aren’t as hairy as the trigonometric ones imo, but still good to know so you don’t have to relearn them later.

Make sure you are comfortable with algebraically changing expressions from one form to another. Factoring and reducing expressions will be super important.

If you have a hard time with any of these, it’s ok; you can review them! If you find that you have forgotten anything you need during your course, see if you can find some excercises in it online or in a book, and do a few so that you are comfortable with it.

Practice

AP Calc involves some proofs, but most of the course is about learning how to do specific types of operations. The best way to prepare is to just do the problems you are assigned for homework, then do more as time passes or if you have a hard time with a particular one.

Memorize formulas as they are introduced. Review them often. Do problems with them.

If you do not understand a concept:

Try to break down why. Do you understand part of it? Write down what you know. See what it is that is stopping you.

Try drawing a picture. Label it. See if you can relate your problem to the visual geometry.

Try working a problem. See where you stop understanding it. Ask yourself why you are doing each step. See if you can explain to yourself.

Look at a worked problem. Explain each step to yourself. See where you stop understanding.

If there a proof involved? Work through the proof, making sure you understand each step. This can give you a solid foundation.

Go to your teacher or a friend with specific questions.

The FRQs and MCQs from previous tests are a goldmine. Do every one you can get your hands on. For FRQs, compare your answers to the model answers given on the College Board website. Mark everything you do wrong. Try to remember it and do it right next time you do a similar problem.

FRQs are great because they tend to incorporate multiple concepts, giving you practice, and they also follow similar patterns. Getting used to those patterns is really helpful.

The Test

Do some full practice tests. Time yourself. Note the concepts you get wrong and review them. Ask someone about things that give you trouble.

Make sure you know all your formulas well.

Make sure you can do everything you will need to with your calculator.

Part of the test is no-calculator. Make sure you can do the sort of problems which appear there without your calculator.

When you take the test:

Sleep.

0/10 do not recommend late night cramming the night or two before the test.

Change your calculator’s batteries. Just so you’re certain it won’t die on you.

Have something to drink on you.

On the MCQs, skip problems you can't do quickly and come back to them. I recommend:

Doing all the easy problems first. The ones that you get instantly. Just read the rest.

Come back and do the ones you need some time for. Ignore any if you have no idea how start or take a lot of time.

Come back for these on the third pass.

They’re all worth the same amount, so don’t worry about specific ones; just get as many as you can right.

Show. You. Work. On the FRQs. Write down everything you can.

If you don’t know how to do the first part of a problem, but the second part relies on it, just pick a number you think is reasonable for the answer to the first part, and use it. You can still get credit for the second part if you use that number correctly.

Don’t stress out too much. Even if you feel terribly, it is quite possible that you did will.

For illustration, I took BC, and I literally cried after the test, because I thought I did terribly. I got a five. The percentage you need to get right to do well is low, and how you feel does not predict how you do.

Take a bit of time for yourself afterwards. It’s going to be May. The weather will be beautiful. Breathe it in. :)

my posts on:

ap in general

ap english literature

ap us history

[mathematicianadda]

A Compendium of Cool Internet Math Things

Here’s an experience I’ve had roughly six million times.

A mathematical topic arises.

“You know,” I say, “someone has a great tweet about this… somewhere…”

In order to find it, I am forced to read all of the tweets, ever.

I am reminded that “all of the tweets ever” is rather too many tweets.

So about a year ago I started a compendium. Tweets, yes, but also videos, apps, memes… anything stimulating or arresting that I can use to embroider my lessons. For a while, this document lived where all important documents live: as a gmail draft. But now I share it as a blog post, and I intend to continue updating it as new ones cross my ken.

NOTE: I will, where convenient, use screenshots and links, because WordPress’s embedded tweets sometimes take ages to load.

  NUMBER AND SCALE

A very strange pricing scheme:

A brilliant anagram from Colin Beveridge:

A gorgeous visualization of prime factors (from this Smithsonian blog post).

The timeless classic Powers of Ten, arguably the best film of 1977 (suck it, Annie Hall):

The mesmerizing interactive “Scale of the Universe” app (which requires you to enable Flash, but just do it).

Also, this black hole:

Science papers are always full of figures, but very rarely are they to scale, but in this astrophysics paper, which I’ve never heard of an astrophysics paper having a figure to scale, the authors included a 1:1 scale of a 5 Earth-mass back hole. Sick as hell. pic.twitter.com/HYhv005gTW

— a small rocket the size of a large rocket (@jaredhead) September 28, 2019

ALGEBRA

Four-story slides shaped like parabolas:

An ellipse as the maximum heights of a family of projectiles:

Throwing an object at the same speed but different angles defines an ellipse via its maximum height https://t.co/vQ8NMssCMf

— 〈 Berger | Dillon 〉 (@InertialObservr) July 22, 2019

Simultaneous equations “in the wild”:

Four place mats, arranged to make a quadratic identity at the dinner table:

Polar coordinates on pizza:

Putting sauce on a pizza. https://t.co/Oe9gsZaSjz

— Machine Pix (@MachinePix) August 28, 2016

GEOMETRY AND TRIGONOMETRY

Volumes of a cylinder, a sphere, and a cone:

Volumes of earth, earth’s air, and earth’s water:

Animated visual proof that any polygon can be rearranged into any other polygon of equal area:

(You’ve just got to click here, it’s amazing.)

For your trigonometric Halloween, the blood function:

Defining a radian with a wooden model:

Tragic Tweet Delete! -- I thought I would at least add it back : ) We are interested in sending these to folks, es… twitter.com/i/web/status/1…

— MathHappens (@MathHappensOrg) October 01, 2019

Simple harmonic motion:

Beautiful shapes created by simple harmonic motion 🧐 https://t.co/ifsFX4nfN9

— Fermat's Library (@fermatslibrary) January 02, 2020

CALCULUS AND DYNAMICAL SYSTEMS

Riemann sums (comparing upper and lower sums as the grid is refined):

Concepts without words: Integration and Riemann Sums bit.ly/2E7iNU3 #math #science #iteachmath #mtbos… twitter.com/i/web/status/1…

— Tungsteno (@74WTungsteno) August 11, 2019

A professor solves an optimization problem (“smallest surface area for a given volume”), writes a company that makes cat food to ask why they don’t use this solution, and receives an incredibly thoughtful and interesting reply:

A real-life butterfly effect:

In office hours, sophomore ⁦@JackSillin⁩ showed me this real world example of the butterfly effect. An unexpectedly… twitter.com/i/web/status/1…

— Steven Strogatz (@stevenstrogatz) September 10, 2019

The exquisite sensitivity of the double pendulum:

50 double pendulums, whose initial velocities differ only by 1 part in 1000 https://t.co/3b75BDkwF1

— 〈 Berger | Dillon 〉 (@InertialObservr) September 30, 2019

PROBABILITY AND STATISTICS

Independence is a delicate and rare phenomenon:

What do probabilistic words really mean?

I see you Anscombe’s Quartet, and I raise you the Datasaurus:

And of course no thread like this would be complete without the datasaurus!

All of these clusters, including the dinosaur, have the same X- and Y-Mean, standard deviation and correlation to two decimal digits. (src: https://t.co/ppLwxtAxOo) pic.twitter.com/7sV8tHCc5m

— λTotoro (@lambdatotoro) January 29, 2019

A delightful game aptly called Guess the Correlation:

The dangers of using r^2 as an effect size estimate:

Why you shouldn’t use r^2 as an effect size estimate – it tells you a dime is “worth” 4 times as much as a nickel. Dan Ozer has one question if you insist on using r^2: “Want to make some change?” #arp2019 pic.twitter.com/okszDQXjtk

— Sanjay Srivastava (@hardsci) June 28, 2019

The normal distribution in action:

The thing I will miss most about teaching is not getting to use this pic of a weight machine to teach normal distribution pic.twitter.com/jNaChtm5tb

— Creosote, King of the Tar Distillates (@edburmila) July 6, 2019

MISCELLANEOUS

Moore’s Law, and the glorious improbability of that exponential growth:

Fascinating: Moore’s Law predictions vs actual growth in transistor count. by @datagrapha reddit.com/r/dataisbeauti… https://t.co/ZwN1dBGE1n

— Lionel Page (@page_eco) September 03, 2019

Quick sort, in an image:

Centrifugal force to restore a whiteboard marker:

I had no idea you could do this to take a dead whiteboard marker and give it life again! Source:… twitter.com/i/web/status/1…

— Robert Kaplinsky (@robertkaplinsky) July 06, 2019

Voronoi diagrams (i.e., which national park is closest to you?):

Set theory (specifically, the power set), where each rectangle is one of the possible sets of these 4 elements (ranging from the empty set in the middle, to the set of all four):

Mathematics in nature:

Stumbled into reading about hermit crabs and wut: https://t.co/ERdEs8TwUK

— Derrick (@geekandahalf) November 29, 2019

from Math with Bad Drawings https://ift.tt/3ajXWeS from Blogger https://ift.tt/3apHWYI

[wodenscild]

Proving De Moivre's Theorem

Building on: Deriving the trigonometric identities

Leads to: The general exact trigonometric forms for 45, 22.5, 11.125, etc.

The Theorem

De Moivre's Theorem is an important theorem when it comes to working with complex numbers, as it serves to greatly simplify otherwise painful calculations. The theorem states:

Where r is the modulus/magnitude of the complex number, and theta is the argument of the complex number. It'll be proven here, using mathematical induction...

The Proof

For the first step, we will prove that our statement is true for the initial value, n=2. This isn't too hard:

Thus, as the left and right hand sides are equal, De Moivre's Theorem is true for n=2.

The next step is to assume that the Theorem is true for some arbitrary value k (which is real and greater or equal to 2*). So we are assuming that the following is true:

The final step is to now show that the theorem is true for n=k+1.

It can get pretty messy, but as it can be seen, with the use of our trigonometric identities, they simplify down nicely to make the left and right and sides equal.

Now, as we were assuming that the Theorem is true for some n=k, which is a real number greater than or equal to 2, then we showed it must be true for n=k+1. The statement has been shown true for n=2, which is a real number equal to 2. 2 is a possible value for k, thus the Theorem is true for n=2+1=3. 3 satisfies the requirements of k, so it is true for n=3+1=4, and so on. It follows from this that De Moivre's Theorem is true for any integer n greater than 2.

Important Notes

I feel like there is an important note to end on... this proof only works with integer powers. The reason for this is the Fundamental Theorem of Algebra, which states that for any polynomial of nth degree, or any number taken to the nth root, there are n solutions. This can be seen in the real numbers, where taking the square root gives 2 solutions. On the complex plane, a point z, with a cube root taken will have 3 solutions (n=1/3), a quartic root will give 4 solutions (n=1/4), a quintic root will give 5 solutions (n=1/5), and so on. De Moivre's theorem still works with these values, but just note that you will only yield one solution if you don't do anything more here.

P.S. Made a little mistake, for the proof, neither n nor k have to be greater than or equal to 2. They can be greater than 0 and still yield the same proof. My bad adkjfdhl

[80daysofproductivity]

HL MATH: Was doing some question bank question for CORE 3 (TRIG) but realized I’ve basically forgotten everything since most of the stuff was done in grade 10 and first semester of hl math - so decided to go back and make some notes and review some old notes too! 

Trig is probably one of my least favourite topics (along with the CORE 1 & 2 too LOOL). Anywho, hoping to get it all done today! Since I’m ahead of my schedule for economics, I decided to take a break from it today and just stick to math!

And for CORE 3 Here’s the syllabus stuff!!!

3.1 The circle: radian measure of angles. Length of an arc; area of a sector.

3.2 Definition of cosx, sinx, and tanx and  in terms of the unit circle. Exact values of sin, cos and tan of 0,π/6, π/4, π/3, π/2 and their multiples. Definition of the reciprocal trigonometric ratios secx, cotx, and cscx  and  the Pythagorean identities:  .

3.3 Compound angle identities. Double angle identities. Not required: Proof of compound angle identities.

3.4 Composite functions and Applications.

3.5 The inverse functions  arcsin, arccos, and arctan ; their domains and ranges; their graphs.

3.6 Algebraic and graphical methods of solving trigonometric equations in a finite interval, including the use of trigonometric identities and factorization. Not required: The general solution of trigonometric equations.

3.7 The cosine rule. The sine rule including the ambiguous case. Area of a triangle as 1/2 (absinC). Applications.

[sanammma]

5 Things

Thank you @the-70s-source for being nice enough to tag me <3 that was really lovely 

5 things you’ll find in my bag:

-A pack of gum -Tissues  -Housekeys -Headphones -Some change

5 things in my room:

-Broken sunglasses that I have to fix but my lazy ass won’t get to do it -Random pieces of paper that I probably won’t ever need but I still keep just in case  -A toy skeleton  -A very messy bed  -My old hamster’s cage 

5 things I’ve always wanted to do in my life:

-Be friends with many people -Be a person who reads ( a lot) -Write a book -Learn to play an instrument  -Keep a box of memories and stuff that I’ll enjoy looking at as an old lady

5 things that I’m currently into:

-Greek Mythology -Old music -Kids and their way of thinking  -The beautiful world of mathematics  -Space 

5 things on my to-do list:

-Jimmy page -Finish the book I’m reading (The broken wings) -Watch the videos about trigonometric identities proofs -Organize files on my computer -Fix my broken sunglasses (probably won’t until my eyes burn )

5 things you may not know about me:

-I like reading history  -I watch anime -I have snapchat and I only put stuff about random instects I see on the street. -I hate makeup but love girls who wear makeup -I am very very very short ..  

[alpha-hsc-maths-tutors-blog]

HSC Maths Extension 1 3U Maths Tutors Sydney Year 11 12​​

Specialist HSC Maths Physics Tutors https://hsc.alpha-maths-tutors.com.au/sydney-specialist-hsc-maths-physics-tutors/hsc-maths-extension-1-3u-maths-tutors-sydney-year-11-12/ Year 11 HSC Maths Extension 1 (3U Maths) Syllabus Working Mathematically Skills Understanding, Fluency and Communicating Problem Solving, Reasoning and Justification Functions ME-F1 Further Work with Functions ME-F2 Polynomials Trigonometric Functions ME-T1 Inverse Trigonometric Functions ME-T2 Further Trigonometric Identities Calculus ME-C1 Rates of Change Combinatorics ME-A1 Working with Combinatorics Year 12 HSC Maths Extension 1 (3U Maths) Syllabus Proof ME-P1 Proof by Mathematical Induction Vectors ME-V1 Introduction to Vectors Trigonometric Functions ME-T3 Trigonometric Equations Calculus ME-C2 Further Calculus Skills ME-C3 Applications of Calculus Statistical Analysis ME-S1 The Binomial Distribution

[mathsodology]

Compositions of trigonometric and inverse trigonometric functions (33)

Compositions of trigonometric and inverse trigonometric functions (33)

Proof of (33). Method 1. We can take the reciprocal of (15): Method 2. Let where and , i.e. What we want is in terms of , so our strategy is to express in terms of using trigonometric identities.