#Trigonometric Formulas Table

[sathcreation]

Expert IIT Mathematics Teacher at Gritty Tech

Introduction to IIT JEE Mathematics Excellence

The journey to cracking the prestigious IIT JEE begins with mastering one of its most challenging components – Mathematics. At Gritty Tech, we understand the intensity and depth required to succeed in this subject. That's why we offer specialized coaching under the guidance of expert IIT Mathematics teachers who not only know the syllabus but also understand the psychology and strategy behind high-stakes exams For More…

Our teaching is not just about covering topics. It’s about building a solid mathematical foundation, developing analytical thinking, and fostering a problem-solving mindset.

Why Expert Teachers Make a Difference

IIT JEE Mathematics is not about rote learning. It demands creativity, speed, accuracy, and deep conceptual understanding. An expert IIT Maths teacher brings all these elements to the table. At Gritty Tech, our instructors are selected through a rigorous process. Each of them has years of experience in mentoring toppers, with proven results across multiple batches.

Key Attributes of Our Expert Maths Faculty:

In-depth Subject Knowledge: Every teacher at Gritty Tech has a mastery over the entire JEE syllabus including Algebra, Calculus, Coordinate Geometry, Trigonometry, Probability, and more.

Strategic Teaching Methods: They know how to break down complex problems into simpler steps and highlight shortcuts that save time in exams.

Focus on Conceptual Clarity: Instead of encouraging memorization, our approach ensures students understand the 'why' behind every formula.

Real Exam Insight: Teachers share insights from previous year papers, trends, and changing patterns in JEE to keep students one step ahead.

Individual Mentorship: Every student is different. Our experts provide personal guidance to overcome weaknesses and build on strengths.

A Curriculum Built for Success

At Gritty Tech, the Mathematics curriculum is not generic. It is crafted specifically for IIT JEE aspirants, keeping in mind the latest trends, difficulty levels, and expected question formats. Our curriculum is reviewed and refined every year by a panel of experts.

The Core Topics Covered Include:

Algebra: Quadratic equations, Complex numbers, Matrices & Determinants, Permutations & Combinations

Calculus: Limits, Continuity, Differentiability, Application of Derivatives, Integrals, Differential Equations

Coordinate Geometry: Straight lines, Circles, Conic Sections

Trigonometry: Identities, Equations, Inverse Trigonometric Functions

Vectors and 3D Geometry

Probability and Statistics

Each topic is taught from the ground up, beginning with theory and moving into solved examples, practice sheets, test series, and revision strategies.

Teaching Methodology at Gritty Tech

What truly sets Gritty Tech apart is not just the content we teach, but how we teach it.

Interactive Concept Classes

Our sessions are designed to be interactive. Teachers use smart boards, real-life analogies, and visual aids to explain abstract mathematical ideas. Every class ends with a set of practice problems that help reinforce the day’s learning.

Doubt Resolution in Real-Time

Students never have to wait till the end of the week to clear their doubts. With dedicated doubt-solving sessions and availability of the faculty even outside class hours, learning continues seamlessly.

Regular Testing and Feedback

Frequent quizzes, weekly tests, and full-length mock exams are conducted to ensure students are exam-ready. After each test, teachers provide personalized feedback and performance analysis.

Adaptive Learning

Not every student learns the same way. Our experts use adaptive learning techniques to identify the learning style of each student and modify their approach accordingly.

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Class 12 Mathematics Notes on Methods of Integration-Comprehensive Study Material

1. Introduction to Integration:

Integration is the reverse process of differentiation, and it helps in finding the area under curves. There are two types of integrals: indefinite and definite integrals. Indefinite integrals provide a general form of antiderivatives, while definite integrals compute the exact area between curves and the x-axis.

2. Methods of Integration:

a. Integration by Substitution:

This method simplifies integrals by substituting part of the integrand. The formula is:

If ∫f(g(x))g'(x) dx = ∫f(u) du, where u = g(x)

Example:

Evaluate ∫(2x)/(x² + 1) dx.

Solution: Let u = x² + 1, then du = 2x dx. The integral becomes ∫1/u du = ln|u| + C = ln|x² + 1| + C.

b. Integration by Parts:

Used for integrals involving products of functions, based on: ∫u dv = uv - ∫v du.

READ ALSO: Class 12 Mathematics Notes: Integration as an Inverse Process of Differentiation

Example:

Evaluate ∫x e^x dx.

Solution: Let u = x, dv = e^x dx. Then du = dx, and v = e^x.

∫x e^x dx = x e^x - ∫e^x dx = x e^x - e^x + C.

c. Integration by Partial Fractions:

Used to decompose rational functions into simpler fractions.

Example:

Evaluate ∫1/(x² - x - 2) dx.

Solution: Factor denominator: x² - x - 2 = (x - 2)(x + 1).

Partial fraction decomposition: 1/(x² - x - 2) = A/(x - 2) + B/(x + 1). Solve for A and B.

d. Integration of Trigonometric Functions:

Useful formulas for integrating trigonometric functions include:

∫sin²x dx = (x/2) - (sin(2x)/4) + C, ∫cos²x dx = (x/2) + (sin(2x)/4) + C.

Example:

Evaluate ∫sin²x dx.

3. Definite Integrals:

Definite integrals calculate the area under a curve within limits [a, b].

Example:

Evaluate ∫[0,1] (x² + 1) dx.

Solution: ∫[0,1] (x² + 1) dx = [(x³/3) + x] from 0 to 1 = (1/3 + 1) - (0 + 0) = 4/3.

4. Properties of Definite Integrals:

Important properties:

1. ∫[a,b] f(x) dx = -∫[b,a] f(x) dx

2. ∫[a,a] f(x) dx = 0

3. ∫[a,b] [f(x) ± g(x)] dx = ∫[a,b] f(x) dx ± ∫[a,b] g(x) dx

5. Practice Problems:

1. ∫(3x + 2)/(x² + 4) dx

2. ∫x ln(x) dx

3. ∫sec²x dx

4. ∫1/(x² + 4x + 5) dx

5. ∫[1,2] (x³ - x) dx

Conclusion:

Integration is a fundamental concept in calculus, playing a vital role in various fields such as physics, engineering, and economics. Mastering different methods of integration—such as substitution, parts, partial fractions, and trigonometric integrals—provides a comprehensive toolkit for solving complex problems.

By practicing both definite and indefinite integrals, students gain a deeper understanding of mathematical principles and their practical applications. Remember, consistent practice is key to excelling in this topic.

Table of Contents:

1. Introduction to Integration

2. Methods of Integration

   a. Integration by Substitution

   b. Integration by Parts

   c. Integration by Partial Fractions

   d. Integration of Trigonometric Functions

3. Definite Integrals

4. Properties of Definite Integrals

5. Practice Problems

[cergisoft]

What is the Tan2x Formula

In trigonometry, the Tan2x formula is a double-angle identity that is particularly useful in solving complex trigonometric problems. The formula is an extension of the basic tangent function and is used to express the tangent of twice an angle in terms of the tangent of the angle itself. Understanding What is the Tan2x Formula can simplify calculations in various mathematical fields, including geometry, engineering, and physics.

The Tan2x Formula is as follows:

Tan(2x) = 2 ⋅ Tan(x) 1 − Tan 2 ( 𝑥 ) Tan(2x)= 1−Tan 2 (x) 2⋅Tan(x) ​ This equation shows the relationship between the tangent of an angle and the tangent of its double, allowing for easier computation of trigonometric values.

How to Use the Tan2x Formula: A Step-by-Step Guide

Step 1: Identify the Angle (x) Before applying the Tan2x formula, you need to know the angle 𝑥 x whose double angle you wish to calculate. This angle could be given in degrees or radians, depending on the problem.

Step 2: Calculate Tan(x) Next, find the tangent of the angle 𝑥 x. You can use a calculator or refer to trigonometric tables to get the value of Tan(x) Tan(x).

Step 3: Substitute the Value into the Formula Once you have Tan(x) Tan(x), substitute it into the Tan2x formula:

Tan(2x) = 2 ⋅ Tan(x) 1 − Tan 2 ( 𝑥 ) Tan(2x)= 1−Tan 2 (x) 2⋅Tan(x) ​ Step 4: Simplify the Expression Simplify the expression by performing the arithmetic operations. This will give you the value of Tan(2x) Tan(2x), which is the tangent of the double angle.

Step 5: Apply in Context Use the result in the context of the problem you’re solving. The Tan2x formula is often used in conjunction with other trigonometric identities, so be sure to consider how it fits into the broader calculation.

3. When should I use the Tan2x formula? You should use the Tan2x formula when you need to find the tangent of a double angle, especially in problems where you’re given the tangent of a single angle and need to compute its double. It’s also useful in solving trigonometric equations and proving other identities.

4. Are there any limitations or special cases for the Tan2x formula? Yes, the formula becomes undefined when Tan(x) = 1 Tan(x)=1 or Tan(x) = − 1 Tan(x)=−1, as the denominator 1 − Tan 2 ( 𝑥 ) 1−Tan 2 (x) would be zero, leading to division by zero. In such cases, the angle 𝑥 x is at 4 5 ∘ 45 ∘ or 13 5 ∘ 135 ∘ (or their equivalents in radians), which are vertical asymptotes for the tangent function.

5. Can the Tan2x formula be used for angles in radians?

Absolutely! The Tan2x formula works for angles measured in both degrees and radians. The same process applies regardless of the unit, so just make sure your calculations are consistent with the units given in the problem.

6. How does the Tan2x graph differ from the Tan(x) graph?

The graph of Tan(2x) Tan(2x) is similar to the graph of Tan(x) Tan(x), but with twice the frequency. This means that the period of Tan(2x) Tan(2x) is half that of Tan(x) Tan(x), so it completes its cycle more quickly. The vertical asymptotes are also more frequent in the Tan(2x) Tan(2x) graph.

Frequently Asked Questions (FAQs):

1. What is the Tan2x formula used for? The Tan2x formula is used to find the tangent of a double angle in terms of the tangent of the original angle. It is particularly useful in solving trigonometric equations, simplifying expressions, and is often applied in fields like geometry, physics, and engineering.

2. How is the Tan2x formula derived? The Tan2x formula is derived using the double-angle identities for sine and cosine 3. Can the Tan2x formula be used with any angle? Yes, the Tan2x formula can be used with any angle, whether measured in degrees or radians. However, you must be cautious with angles where Tan(x) = 1 Tan(x)=1 or Tan(x) = − 1 Tan(x)=−1, as these values make the formula undefined due to division by zero.

4. What happens when Tan(x) = 1 Tan(x)=1 or Tan(x) = − 1 Tan(x)=−1? When Tan(x) = 1 Tan(x)=1 or Tan(x) = − 1 Tan(x)=−1, the denominator of the Tan2x formula becomes zero, which means the formula is undefined. This typically occurs at angles like 4 5 ∘ 45 ∘ or 13 5 ∘ 135 ∘ (or their radian equivalents), where the tangent function has vertical asymptotes.

5. How does the Tan2x graph differ from the Tan(x) graph? The graph of Tan(2x) Tan(2x) differs from the Tan(x) Tan(x) graph in its frequency. The Tan(2x) Tan(2x) graph has twice the frequency, meaning it completes its cycle in half the period of the Tan(x) Tan(x) graph. Additionally, the vertical asymptotes in the Tan(2x) Tan(2x) graph occur more frequently, reflecting the doubled angle.

Conclusion

Understanding how to use the Tan2x formula is essential for solving a variety of trigonometric problems. Whether you’re calculating double angles, simplifying expressions, or solving equations, this formula provides a straightforward method for working with tangent functions. By following the steps outlined in this guide, you can confidently apply the Tan2x formula in your studies or professional work.

Remember, the key to mastering trigonometry lies in practice. Try working through different problems using the Tan2x formula to deepen your understanding and improve your mathematical skills.

[edukotangp]

Must-Study Chapters for JEE: A Comprehensive Guide

Mechanics: The bedrock of physics, covering kinematics, laws of motion, work, energy, power, rotational motion, and gravitation.

Electrodynamics: A conceptually rich section encompassing electrostatics, current electricity, magnetism, and electromagnetic induction.

Modern Physics: Relatively easier with topics like atomic structure, nuclear physics, dual nature of matter, and semiconductors.

Optics: Crucial for engineering applications, covering ray and wave optics.

Thermodynamics and Kinetic Theory: Bridges physics and chemistry, covering laws of thermodynamics, Carnot engine, entropy, and heat transfer.

Chemistry

Physical Chemistry: Numerically oriented, focusing on mole concept, thermodynamics, kinetics, electrochemistry, ionic equilibrium, and solutions.

Inorganic Chemistry: Factual and memory-based, covering periodic table, chemical bonding, coordination compounds, p-block, d-block, and f-block elements.

Organic Chemistry: Concept-heavy, emphasizing general organic chemistry, hydrocarbons, alcohols, phenols, ethers, carbonyl compounds, and amines.

Mathematics

Calculus: The foundation for higher mathematics, covering limits, continuity, differentiability, applications of derivatives, definite and indefinite integrals, and differential equations.

Algebra: Core of JEE math, encompassing quadratic equations, sequences and series, matrices and determinants, complex numbers, and probability.

Coordinate Geometry: Often considered easy, covering straight lines, circles, parabolas, ellipses, and hyperbolas.

Trigonometry: Versatile and applicable across topics, covering trigonometric ratios, identities, heights and distances, and inverse functions.

Vectors and 3D Geometry: Can be challenging but highly scoring with practice, covering vector algebra and three-dimensional geometry.

Tips for Effective Preparation

Understand Concepts: Don't just memorize formulas; grasp the underlying principles.

Practice Regularly: Solve previous year papers and mock tests consistently.

Focus on Weak Areas: Identify and strengthen your weak points.

Revise Consistently: Regular revision is crucial for retention.

Stay Updated: Keep track of exam pattern changes and syllabus modifications.

[tparadox]

People talking about how "oh they're just now starting to teach math by explaining how it works" sound weird to me. I was at school after New Math and before Common Core and they did too explain it to us. Yeah, we got addition and multiplication tables to memorize, but only after they explained to us how addition works with counting demonstrations and addition explanations. Even derivatives and trigonometric proofs, they spent a day on deriving the concepts before giving us the shorthand tools. There are some formulas that are useful at much lower levels than you need to understand why they work, like for the area of a circle and volume of a sphere, but I'm pretty sure when we learned the math necessary to derive them, my teachers would always take a moment to say "by the way, here's why the circumference of a circle is πr²".

[tutoroot]

What are the Standard Angles of Trigonometric Table? 

The Trigonometric Table is essentially a tabular compilation of trigonometric values and ratio for various conventional angles such as 0°, 30°, 45°, 60°, and 90°, often with extra angles such as 180°, 270°, and 360° included. Due to the existence of patterns within trigonometric ratios and even between angles, it is simple to forecast the values of the trigonometry table and to use the table as a reference to compute trigonometric values for many other angles. The sine function, cosine function, tan function, cot function, sec function, and cosec function are trigonometric functions.

The trigonometric table is helpful in a variety of situations. It is required for navigating, research, and architecture. This table was widely utilized in the pre-digital age, even before pocket calculators were available. The table also aided in the creation of the earliest mechanical computing machines. The Fast Fourier Transform (FFT) algorithms are another notable application of trigonometric tables.

Standard Angles of Trigonometric Table

The values of trigonometric ratios for the angles 0°, 30°, 45°, 60°, and 90° are widely utilized to answer trigonometry issues. These values are related to measuring the lengths and angles of a right-angle triangle. Hence, the standard angles in trigonometry are 0°, 30°, 45°, 60°, and 90°. Following are the trigonometry angle tables (show the figure).

The trigonometric table in its simplest sense refers to the collection of values of trigonometric functions of various standard angles including 0°, 30°, 45°, 60°, 90°, along with other angles such as 180°, 270°, and 360°. These angles are all included in the table. This makes it easier to determine and arrive at the values of the trigonometric ratios in a trigonometric table, also, the table can be used as a referral illustration to compute trigonometric values for various other angles, due to the patterns that are seen within the trigonometric ratios and those between angles.

The table as one might note, consists of trigonometric ratios – sine, cosine, tangent, cosecant, secant, and cotangent. The short forms of these are very popular – sin, cos, tan, cosec, sec, and cot, respectively. Always, memorize the values of the trigonometric ratios of the standard angles.

Always remember these points in the trigonometric table:

In a trigonometric table, the trigonometric values for complementary angles, such as 30° and 60° are measured by applying complementary formulas for various trigonometric ratios.

The value for some ratios in a table is ∞ “not defined”. The reason is that while computing values, the denominator shows a “0”, which implies that the trigonometric value cannot be defined, and is as good to be the equivalent of infinity.

Please notice the sign change in the values at places under 180°, and 270°, for values of some trig ratios in a trigonometric table. This happens because there is a change in the quadrant.

Trigonometric values

As explained, if trigonometry deals with the relationship between the sides of a triangle (right-angled triangle) and its angles, then the trigonometric value refers to the values of different ratios, sine, cosine, tangent, secant, cotangent, and cosecant, all in the trigonometric table. All the trigonometric ratios are in relation with the sides of a right-angle triangle. The trigonometric values are derived applying these the ratios. Refer to the following steps to create trigonometric values:

Steps to Create Values for Trigonometry Table

Step 1: 

Make a table with the top row showing the angles such as 0°, 30°, 45°, 60°, and 90°, and the first column listing the trigonometric functions such as sin, cos, tan, cosec, sec, cot.  

Step 2: Determine the value of sin 

To find the sin values, divide 0, 1, 2, 3, 4 by 4 under the root, in that order.

Step 3: Determine the value of cos 

The cos-value is the inverse of the sin angle. To find the value of cos, divide by 4 in the opposite order as sin.

Step 4: Determine the value of tan 

Tan is defined as sin divided by cos. Tan equals sin/cos. Divide the value of sin at 0° by the value of cos at 0° to get the value of tan at 0°.

Step 5: Determine the value of cot 

The reciprocal of tan is the value of cot. Divide 1 by the value of tan at 0° to get the value of cot at 0°. As a result, the value will be as follows: cot 0° = 1/0 = Unlimited or Not Defined  

Step 6: Determine the value of cosec 

The reciprocal of sin at 0° is the value of cosec at 0°. 

Step 7: Determine the value of sec 

Any common values of cos may be used to calculate sec. The value of sec on 0° is the inverse of the value of cos on 0°.

While we learn trigonometric values of the trigonometry tables, it will also be interesting to take note of the application areas of the table. On a broader note, the trigonometric table is used in:

Science, technology, engineering, navigation, science and engineering. Before the advent of the digital era, the trigonometric table was very effective. In the course of time, the table helped in the conceptualization of mechanical computing devices. Trigonometric tables are also used in the Fast Fourier Transform (FFT) algorithms.

Important Tricks to Remember Trigonometry Table

Knowing the trigonometry table can help you answer trigonometry problems and remembering the trigonometry table for normal angles ranging from 0° to 90° is simple. Knowing the trigonometric formulae makes remembering the trigonometry table much easier. The trigonometry formulae are required for the Trigonometry ratios table.

The angle values of trigonometric functions, cotangent, secant, and cosecant are computed by applying these standard angle values of sine, cosecant, and tangent. All the higher angle values of trigonometric functions such as 120°, and 360°, are easier to compute, through the standard angle values in a trigonometric values table.

If you still can’t remember the values of Trigonometry tables, consider Tutoroot personalised sessions. Our Maths online Tuition session will help you clearly understand the table along with tricks to memorize. 

[thepenpostblog]

Trigonometry Formulas

Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. It’s a fundamental topic in mathematics and finds applications in various fields, from physics and engineering to astronomy and navigation. To master trigonometry, one needs to be well-versed in its numerous formulas and identities. In this comprehensive guide, we will explore a wide range of trigonometry formulas, making them accessible and understandable. Whether you’re a student looking to ace your trigonometry exams or a professional seeking to apply trigonometric principles in real-life scenarios, this article has got you covered.

Trigonometry Formulas List:

Let’s kick things off by presenting a list of the fundamental trigonometry formulas that we will delve into in more detail:

Basic Trigonometric Function Formulas

Reciprocal Identities

Trigonometry Table

Periodicity Identities (in Radians)

Cofunction Identities (in Degrees)

Sum & Difference Identities

Double Angle Identities

Triple Angle Identities

Half Angle Identities

Product Identities

Sum to Product Identities

Inverse Trigonometry Formulas

Trigonometry Formulas from Class 10 to Class 12

Trigonometry Formulas in Major Systems

[etcetraetcetra]

•1.06.2023, Thursday•

DAY 1/30

I am thinking about starting a 30 days productivity challenge. (Is that what it is called?) We already reached mid year of 2023 and I m nowhere near completion of syllabus and it’s getting very frustrating. Anyway, today is a Thursday and I don’t have any tuition today. I can study whatever I wish to today :)

Things I did today-

Wrote the remaining theory for Integration

Learnt the formulae for Integration

Completed physics homework

Practiced Inverse Trigonometric Functions

Revised Limits

Watched a lecture on Periodic Table and revised on the topic

Completed Trigonometry from NCERT

[cherrysung]

attention

pairing: nerd!jaemin x reader

genre: smut

warnings: language, unprotected sex (stay safe!), deepthroating, oral (m. receiving), creampie, choking, fingering, dirty talk, slight degradation

prompts: none

summary: innocent and harmless to the eyes of everyone else, the quiet boy sitting next to you couldn’t stand your impatience anymore. maybe another lesson needed to be taught.

requested by @jjuullnnaayyaa.

word count: 2.5k

note: I hope you like this! this is the second part (requested) to my first nerd!jaemin scenario, which you can check out here! also, @glossyjaems have fun lmao

cherrysung’s navigation

“Stop it.”

“Stop what? I’m not doing anything.”

Clenched jaw and threatening, narrowed eyes, Jaemin could only spare your mischievous smile a short glance before turning his attention back to the boring teacher in front of you. She spoke about some physics formulas and confusing projectile motions, yet although your focus seemed to be directed with full interest at her explanations, it was far from that. Instead, while you stared at the white board covered in multiple doodles, your hand rested not so innocently on Jaemin’s thigh. All under the desk and for nobody else to see.

You scribbled down the long formulas on your notebook in an unbothered nature, left hand moving slowly in up and down motions on your desk partner’s thigh, occasionally squeezing and teasing to reach unholy areas. Jaemin appeared to be sucking in ragged breaths, the spectacles sitting on the bridge of his nose slightly sliding down due to the increasing perspiration on his face.

He looked nervous, angered even. Gritting his teeth or biting on his lower lip as your hand gradually moved higher and closer to the growing bulge inside his beige dress pants.

The two of you had been seeing each other for a considerable amount of weeks now, thankful that a simple tutoring session was enough to kick off your current relationship; date after date, you always picked up a new characteristic from his reserved personality, only adding to your belief that really—not all is what it appears to be. Despite the fact Jaemin was in reality a freak and nothing like the nerd he had come to be recognized as, he was still fairly shy, and had struggled big time to ask you to be his girlfriend. Stumbles and stutters had been leaving his lips infinitely until you kissed him and spared him the work, answering his timid question with a simple yes.

His breath abruptly hitched, hands roughly gripping onto the desk’s edge and pencil when you curiously palmed his hardened cock. The muscles on his thighs were flexed, Adam’s Apple bobbing up and down as he forcefully swallowed with nervousness and want.

“Stop that. You know what I’m talking about.” He whispered, eyes still fixated on the class before the two of you whilst using his free hand to grip yours gently and stop it from further movements. “What’s with you? We’re in the middle of class, we need to pay attention or we won’t understand anything afterwards.”

You smiled, admiring his control and obedience. Leaning closer to his ear, you whispered anyways, “but the only one I want to pay attention to is you. Jaemin, I want to fuck you.”

In a bold move, you guided his hand under your plaid skirt and towards your soaked heat, moving your flimsy underwear to the side to run his index finger through your wet folds.

Jaemin cursed under his breath, tongue swiping swiftly on his bottom lip as he lowered his head so the teacher wouldn’t notice the slight pink tint that had flushed warmth into his cheeks. His fingers started moving on their own, thumb rubbing your clit gently before plunging two fingers into your dripping hole. You allowed him to continue the discrete assault on your pussy as you brought your hand back to its previous position—his dick. Experimentally, you unbuttoned his pants with only one hand, taking his hard member out after he had obediently raised his hips to lower down the fabric of his black underwear.

He continued to pump his fingers in and out of your cunt at an average pace, unable to move them any faster due to the little space between you and the desk. Still, he didn’t fail to please you, curling them and hitting your sensitive spot successfully with no trouble whatsoever. Meanwhile, you stroked his length slowly and tightly, using the precum leaking from the tip to coat his dick. All of this while he innocently rested his chin on his hand, mindlessly nodding at whatever the teacher said and as you pretended to listen, now writing random things on your notebook, because you weren’t bothered enough to actually keep scribbling down notes from the subject.

“So, Na Jaemin, what do we do next? Give a brief explanation too.” The teacher spoke up, all heads turning your way and prompting both your movements to halt for a moment. Though, it didn’t take long for them to resume, as if twenty-eight eyes weren’t situated on the boy next to you. If only they knew what was happening underneath the metal table.

Jaemin fixed his glasses, and you could only widen your eyes in annoyance at how fast his demeanor changed, gaze turning shy and cheeks still warm from your ministrations; but, fingers reaching deeper while his thumb returned to caressing your sensitive bub. “You move the number three to the other side of the formula,” he spoke with confidence, hips involuntarily jolting up slightly when your thumb swiped for a little too long on his tip. “The reason why you have to do this, is to actually get the formula working. Otherwise, the answer will be unobtainable.”

“That is correct.” The teacher nodded with a small smile. “Thank you, Na Jaemin. I didn’t expect any less from my best student, anyways.”

For about ten more minutes, the lesson carried on at an extremely boring and deathly pace, the sound of markers coming into contact with the board and quiet complaints from students the only things that could be heard between the four, dull white walls. Jaemin and you had resorted to torturing each other in the form of teasing, unwilling to have any type of release underneath a dirty school desk.

The bell rang, signaling the end of the lesson as the two of you stopped your actions immediately. You swiftly fixed your skirt, and Jaemin made sure to zip his pants back up only after quickly licking his fingers clean from your essence while gazing deeply with desire into your eyes.

“Class, you’re dismissed.”

You stood up a ridiculously desperate pace, only to be pulled back down onto your seat by Jaemin. “What the fuck! Jaemin, I’m horny, let’s leave.” Your threats came out in a whisper, but they were enough to bring a sinister smile to Jaemin’s features.

“Don’t worry, I have a better idea.” He waited for the last pair of students to leave the classroom, watching as the teacher stood with her bag, hands holding onto her homemade lunch. She walked carelessly towards the door, steps coming to a halt upon noticing the two of you were still glued to your seats.

“Guys? It’s lunchtime, why are you still here?”

Jaemin took out his calculus textbook along with his calculator, placing the items on the table before pointing at them and then at you. “Could we stay here? She’s having trouble with Inverse Trigonometric Functions and we have calculus class after lunch break. We thought it’d be a good idea to stay here practicing just to warm up.”

She pondered for a few seconds, before nodding her head. “Sure, but if you get hungry don’t hesitate to go to the cafeteria, okay? It’s always better to avoid an empty stomach. Good luck and work hard!”

With that, she left, softly closing the door after praising the two of you for being such ‘dedicated students’. Jaemin chuckled quietly before locking the classroom door, walking back to the table and pulling you up rapidly, wasting no time in locking lips with you after having to put up with your endless teasing for an entirety of forty-five tedious minutes.

You held tightly onto his shoulders as he walked you backwards until your back hit a wall, taking advantage of the gasp that escaped your lips to kiss you deeper, tongue fighting with your own as his hands roamed with lust all throughout your body. He unbuttoned your blouse just enough to free your tits, pulling your bra down as his fingers pinched your nipples playfully. Jaemin swallowed your moans, turning the both of you around so his back was pressing against the wall now, rubbing your shoulders tenderly before pushing you down to your knees.

“You were so naughty today, weren’t you? I thought you had already gotten a well deserved lesson taught? Seems like you didn’t learn shit, my dirty princess. Should I be harder on you?”

A whimper left you as you kneeled on the cold, tiled flooring, looking up at Jaemin obediently as he lovingly tucked your hair behind your ears. He released his throbbing cock from his undergarments, right hand bringing your locks up into a makeshift ponytail before pushing your head closer to his leaking shaft.

“Do you have make-up removal wipes here?” He simply asked, free hand pumping his member as he looked down at you with hooded eyes.

You nodded.

“Then we’re fine. Open your pretty mouth, doll.”

You listened to his demand right away, mouth opening with your tongue slightly sticking out as Jaemin placed his cock on top of the warm muscle, breathing instantly becoming irregular. In a rapid motion that caught you off guard due to no warning given beforehand, he roughly thrusted his hips forward, dick reaching the back of your throat unexpectedly as you gagged, fighting the urge to cough out as his actions became consecutive and merciless.

He successfully caressed the depths of your throat experimentally, the outlines of his member occasionally showing whenever you were able to avoid the need to gag. His low grunts resonated throughout the empty classroom, fucking your mouth ruthlessly until tears mixed with black mascara were running down your face uncontrollably and your throat burned as it begged for an inexistent break.

His movements only slowed when his dick twitched, hand forcing you down onto his cock ‘till your nose was pressed against his tummy once last time before letting you go, gasps and coughs desperately leaving your lips in an instant as you attempted to catch your breath. Jaemin let out an exhale and pulled you back to your feet once again, swiftly wiping the black tears from your cheeks as he admired your reddened eyes. He wordlessly asked if you were fine, smiling gently when you nodded your head.

“Come here.” He guided you towards the teacher’s desk—again, giving you no time to process what was happening before he had pushed your upper body down, chest and cheek pressed onto the polished wood while his hand rested on your back. He pushed your skirt upwards, the fabric bunching up at your waist as he moved your pink panties to the side.

Your glimmering cunt stared back at his intense gaze, juices now running down your inner thighs after enduring the attack of his fingers with no room for release. He ran the tip of his dick up and down your folds teasingly, breath hitching while whines escaped you.

“Jaemin, please.” You whispered, mind unable to think of anything else but the feeling of his shaft caressing your pussy.

“What, princess? What do you want? Use your words.”

“Please, fuck me. Fuck me hard, so hard I won’t be able to focus the rest of the lessons. Fuck me so hard I will keep my hands to myself during class next time I feel like a little slut.”

“Good choice and wording, love. We only have about… ten minutes left before lunch break ends, so let’s make this worth it. Let me fuck the whore out of you, alright?”

No more words said, he rammed into you, setting a ferocious and reckless pace right away as your frail body lunged forward weakly with every fierce thrust that he delivered. The tip of his dick kissed at your sweet spot every time, eliciting gasps and whimpers from you that only served to help Jaemin increase his already inhumane speed.

“You like that right?” He gritted out. “This is what you wanted all along, right? You were so desperate for my dick that you couldn’t wait until we got home; no, you just had to touch me in the middle of a lesson. What a dirty doll you are.”

He continued to pounce into you, thrusts never decreasing in rhythm as he pulled you up close to his chest by your hair, free hand circling around your neck before squeezing lightly. You let out a particularly loud sound of pleasure, hands shooting up to rest on Jaemin’s much bigger one, silently begging him to keep the hold on your neck.

A surprised chuckle left his lips, hand squeezing once again as your walls clenched around him, tightly engulfing his dick deliciously. He moaned, head falling to your shoulder before biting gently onto the skin, sucking and then soothing the burning feeling with a slight swipe of his wet tongue. Whispering dirty things into your ear, you whined as his words swirled chaotically in your mind; promises of what more he’d do to you once you were alone and praises of how good you felt drowning you.

You felt his member twitch inside of you, thrusts gradually becoming sloppier by the moment. The grip on your neck became stronger, nearly preventing you of all ability to breath.

“J-Jaemin, I’m close!”

“Yes, princess, go ahead and let go. I want to feel you. Go ahead and cum all over my dick.”

Jaemin softly nibbled on your earlobe, the action enough to bring you to your climax. Your cunt clenched unstoppably around his shaft, juices mixing along with his as he reached his release as well, cum filling your walls to the brim through white, warm strings. His hips abruptly halted, dick still buried deep inside you while the two of you attempted to catch each other’s breaths.

Upon the loud sound of the bell ringing through the classroom and school halls, you both fixed yourselves in record time. After zipping up his pants, Jaemin helped you button up your shirt, fingers collecting his juices that had trickled down from your pussy before inserting the milky substance inside of you once again and arranging your underwear back in place, your legs shaking at the oversensitivity. He smirked and cupped your heat, eyes twinkling at the squeal that left your lips. “Keep that in you until we return back to my house.”

Quickly, he was able to unlock the door on time as the two of you now sat at your assigned desk, while Jaemin softly wiped off the black stains of mascara from your cheeks.

“Oh my God, what happened? Why is she crying?” The teacher asked worriedly the moment she stepped into the classroom.

“Oh, don’t worry,” Jaemin smiled sweetly at the elderly woman, “she became angry because she couldn’t get the topic, so angry that she started crying.”

“Oh, sweetie,” she smiled at you with pity, “you’ll get it soon! Don’t be too hard on yourself.”

“I know right! I told her the same thing.” Jaemin giggled, aware of the strong glare you sent his way. “Don’t worry, though, I promised her I will keep teaching her the lesson after school.” He smiled innocently, eliciting coos from the teacher who had been unaware of the slightly harsh pat that he delivered to your sensitive heat.

You gulped, rubbing your thighs together at the feeling of his cum still sitting in the depths of your walls, and the intrigue of what else he would ‘teach’ you behind closed doors.

[onlinelearningcourseguide]

NCERT Solutions For Class 10 Maths Chapter 8

Class 10 maths chapter 8 is Introduction to Trigonometry. Trigonometry is one of the important and scoring chapter of class 10. But most of the children face problems during solving the trigonometric equation and learning the formulas. Reminding the trigonometry formula always be the trouble, but there are some simple tips to solve the trigonometry formulas.

Tip 1) Always Start from the More Complex Side.

Tip 2) Express everything into Sine and Cosine.

Tip 3) Combine Terms into a Single Fraction.

Tip 4) Use Pythagorean Identities to transform between sin²x and cos²x.

Tip 5) Know when to Apply Double Angle Formula (DAF)

Topic Names of Chapter 8

8.1 Introduction

8.2 Trigonometric Ratios

8.3 Trigonometric Ratios Of Some Specific Angles

8.4 Trigonometric Ratios Of Complementary Angles

8.5 Trigonometric Identities

8.6 Summary

Who is known as Father of Trigonometry

The first known table of chords was produced by the Greek mathematician Hipparchus in about 140 BC. Although these tables have not survived, it is claimed that twelve books of tables of chords were written by Hipparchus. This makes Hipparchus the founder of trigonometry.

Trigonometry Formulas

Here are some important trigonometry formulas for class 10 chapter 8, which will definitely work during the time of exams. The formulas chart is given below-

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[ask-de-writer]

KURIN’S FOLLY : World of Sea : Part 13 of 15

KURIN’S FOLLY

Part 13

by

De Writer (Glen Ten-Eyck)

23,699 words

© 2020 by Glen Ten-Eyck

writing begun  2006

All rights reserved.

Reproduction in any form, physical, electronic or digital is prohibited without the express consent of the author.

//////////////

Copyright fair use rules for Tumblr users

Users of Tumblr.com are specifically granted the following rights.  They may reblog the story provided that all author and copyright information remains intact.  They may use the characters or original characters in my settings for fan fiction, fan art works, cosplay, or fan musical compositions.

All sorts of fan art, cosplay, music or fiction is actively encouraged.

///////////////////////

New to the story?  Read from the beginning.  Part 1 is here

///////////////////////

The whole crowd was buzzing with amazement.  This was not the Juris that people knew.

The Council convened shortly after Kurin announced her choice for head of the boat shop.  Juris showed up, acknowledging that he had no formal standing to be present, he nonetheless asked for and was granted permission to address the Council.

After Conner Palin Longin was enrolled in the Council as head of the boat shop, Juris requested a chance to speak.  Kurin seconded his request.

Gravely, Juris stood and gazed nervously at the assembled Masters for a moment before speaking. These were friends.  Friends that he had put into a terribly difficult position.  He knew it.  They knew it, too.  

They just didn’t know how difficult it was going to get.  Uneasily he gripped his hands and wrung them as he spoke.  “My friends, I have a difficult task to do.  First, before anything else, I must apologize for the problems that I have caused for this ship and my beloved Craft.  

“That said, I know, now that it is too late, that I have been so far onto dry land that I could not even tell that I was aground.  Our law accepts the declaration of sanity from any individual who is competent in his work.  I am asking that you accept not only an apology but that you amend the declaration of sanity in my case.  I have only in the last few days come to understand the nature of my madness and how shallow the water I was in.  Can you accept this?”

Looking for support among the assembled Masters, he saw two things.  One stung like the venom of a jelly-floater.  It was the nodding sympathy of his friends.  The other was almost worse.  They were granting his request.  Sour at the sympathy but with hidden jubilation, he realized that he’d pulled it off!  The show of hands was only a formality.

It was necessary to arrange for the classes in navigation and mapping to begin.  Ships were paying good skins for the service.  It would not do to disappoint them.

Under clear skies, with only a few clouds, and a scouting sweep by High Cloud to assure that the next few days would stay fair, the catamarans were launched. Students and instructors boarded and sails were raised.  Fast as the Longin was, the little cats were swifter.  The sails of the Longin were out of sight over the horizon in only a few hours.

At a good location for what she wanted to teach, where the waves were pronounced and simple to work with, Kurin had the boats assemble.  When they were all fastened together by painters, so that nobody could drift out of hearing, Kurin sat herself comfortably on the cabin roof of her boat and began to lecture on the nature of the waves and what it was that they were trying to feel.

Soon the instructors from the Longin took over on the basics in each boat except Kurin’s.  There, a separate and much smaller group were beginning the basics of bottom mapping.

Two of this class were Licensed First Officers.  The other three were young, no more than twelve to fifteen Gatherings old.  The younger children were, as was often the case, far better at sensing the waves and their changes.

Each student had a trigonometric abacus, a mapping tallow slate and a slate on which to do their work. At the moment, they were listening to Kurin’s explanation of a problem.

“We have three intersecting waves here.  The one we need to concentrate on is the lowest one.  It is also spaced the longest.  Try to feel its period.  When you think that you have it, time it three times with the precision hand of the water clock.  Write the times down.  If any of them differs by more than one second, you have made an error.  Try again.

“This wave is generated by the lift and fall of the whole ocean, due to the primary tidal bulge.  In deep waters, this wave will tell us the depth with a bit of figuring. I have set up the formulas that you will use, in the order of use on my big slate.  Make a note of them.  You will need these formulas a lot.”

The students crowded around her table and copied the formulas.  One of the officers gestured at the big tallow slate and asked, “How do we get this number here?  I don’t understand.”

Kurin settled back and asked, “What things do we know about the wave?  Anyone?”  She pointed, “Markham?”

One of the teenagers responded, “We know its period, in seconds.  We can measure its amplitude. You said in lecture, that there is a relationship between amplitude and crest to crest time, that gives the speed of the wave.  Speed and time give us the length.  There was something that I didn’t understand about triangle functions, too.”

Kurin nodded, “Right.  With the data that you described, we can create a triangle that is similar to the one created by the wave.  There is an easily calculated proportion that will then give you the depth, accurate to within a few fathoms in water of this depth.  It works like so.”  She began to draw on her big tallow slate, illustrating the points of the problem as she progressed.

By the end of the lesson, as the Longin hove into sight on the southern horizon, everybody was getting a handle on the tricky work.  The small boats, with their deliberately flexible design, allowed even the people who normally could not feel the waves to become aware of them.  The system was working better than Kurin had hoped.

No sooner had she climbed the cargo net strung over the side as a ladder, back to the ship’s deck than Lissa took her aside.  Urgently, she said, “Kurin!  They’ve closed your shop and put a Council seal on the door!  You are accused of violating the innovation laws.  I think that Juris is behind it. He and Yoram were asking a lot of questions about our work just before it happened.”

Kurin simply asked, “How long have you been shut down?”

“Since ten this morning.”

Kurin nodded to herself.  Then she shook off Lissa and set out to find Alor.  Down in Alor’s cabin, Kurin sat and said bluntly, “Remove the Council seal on my shop and name the complainant.  If it is the Council of the Longin who is Complainant, I will have to file a legal action and that will dissolve the Wergeld.”

Alor looked tiredly at Kurin and said, “Not if the complaints have substance.  We have found that you are using at least five Proprietary Innovations in that shop and are paying license on only one.”

Kurin shrugged.  “Perfectly true,” she stated.  “I pay the Weavers Guild for the screen printing process, the same as any who use it.  As for the others, have you looked up the holders of the Innovations to see who I should be paying?”

“Not yet,” answered Alor. “Yoram just brought the matter to the attention of the Council. There were so many violations found on questioning your helpers, that we sealed the shop first.”

“You should have looked first,” said Kurin, “especially since it was Yoram, Juris’ crony, who stirred things up.”  She pulled the massive Current Innovations of the Naral Fleet from the shelf and began flipping the pages expertly.  Under “Cooks Guild, she found and pointed out, A method for using the Public Innovation Mess Cleaner to produce etched designs in many materials. Only a bit further on she showed Alor, A system for making many identical pieces.

Then she went the Boatbuilders Guild section and found An engine for geometrically laying out accurate radii.  This was followed by An engine for dividing any length into equal parts.  

In the Ropers Guild section she pointed out A new core for cable of any Twist or Weave.  Then in the Held Through No Guild section, she showed Alor A device to laminate very thin sections and A method to produce Glue of great flexibility and softness.  There was also A double escapement for precision mechanisms requiring an accurate rotational rate.

Kurin put the tome down and said, “That about covers it for my shop.  I don’t pay for the use of those Innovations because they’re mine.  For at least two of them, Juris knows that I hold them.

“I think that he knows that he is going down and is trying to drag you under as well.  Perhaps he thinks that I won’t strike at a ship that I love to get at him.  If that’s the case, he’s right.  Dear Dragons, I don’t even want to see him lost!  I just can’t stop it!”  She broke down and began to cry pounding her fists on the big book in frustration.  From a long habit of avoidance, not from lack of feeling, Alor sat helpless and watched.

Lissa had followed Kurin and listened from the door.  She gathered the weeping girl into her arms and rocked her.  Lissa stroked Kurin’s hair and crooned meaningless comforting sounds until Kurin got her feelings back under control.

As she held Kurin close and listened to her efforts to settle down, Lissa said perceptively, “That memory of yours may be a Dragon’s Gift but I have to wonder sometimes if Blind Mecat really understood what she was doing to you. Not being able to forget seems to be like trying to fight a Kraken Hag all by yourself.”

Bleakly, Kurin replied, “Sometimes I’d rather.  A Kraken Hag is simple.  It just wants to eat you.  This business with Juris is giving me nightmares like I haven’t had since father died.”

Somewhat helplessly, Lissa patted Kurin’s back and started to say, “I’m sorry abou . . .”

Kurin placed a finger on her mother’s lips and whispered, “That’s past and forgiven.  You just wanted father back even more than I did.  More than anyone.”

Turning in her mother’s arms but not trying to be free of them, Kurin asked, “Now, about our shop and Juris.  What shall we do, Alor?”

Firmly, Alor replied, “First, we unseal your shop.  Then, we get the Masters, publicly, over dinner, to fix a fine against Juris’ account.  Legally, he is no longer responsible and cannot be made to pay but we can vote to put him off the ship as a danger if he does not.  

“Frankly, I can see no other way to end his devious meddling in the running of this ship.  He will not keep his word any longer.  This has destroyed what little trust I had left in him.  I think that we need to search his things to see if there is more to his actions than appears on the surface.”

At dinner, in front of the entire ship, except for those on duty, Juris and Yoram were confronted with his attack on Kurin’s shop.

Almost nonchalantly Juris replied to the charge, “I simply had attention called to what appeared to be irregularities in the conduct of Kurin’s business.  It was the Council that acted inappropriately.”

Alor asked grimly, “Are you aware of the limitations on your privacy rights resulting from your insanity declaration?”

Juris hesitated.  “What does that have to do with anything?”

Alor responded, “We found this in your things.  Would you care to comment?”  She was holding a tallow slate that had been erased by rubbing over the surface.  It had been carefully scraped to restore the original text.

Juris didn’t do more than glance at it.  “A list,” he snorted, “A mere to-do list.”

Captain Mord responded to that by calmly reading, “One: see to the use of Glue formula #17 on Kurin’s boat.  Two: get Kurin to help me with insanity plea.  Three: complain about Kurin’s shop to close it.  Four: prepare Kurin’s funeral speech.”

He cocked his head at Juris and stated, “This shows that you do not meet the legal test of insanity.  We checked on the things in this list.  Glue #17 is a water soluble formula used for temporary assemblies.  You have followed through on items One Two and Three or they would not be rubbed out.  We have someone looking over the small boats now for the presence of #17 Glue.”

Kurin recoiled in shock.  She dumped over her bench in her rush to leave the mess.  Lissa was after her like a harpoon shot from a catapult.  She found Kurin leaning over the rail, vomiting.

Lissa took Kurin’s shoulders and held her until her stomach was empty.  Sympathetically she asked, “What set that off, Kurin?”

“Mass murder,” said Kurin shakily.  “Juris couldn’t know which boat I would be on.  He has to have sabotaged all the contest boats.  The entire navigation class and all of our instructors are in danger.  Whoever mixed the Glue for the boats was in on it.  #17 stinks when you first put in the soap. After that it behaves normally.”

They met the inspection party as they were returning to the mess.  “What did you find?” asked Lissa curiously.

“First boat we looked at,” replied the leader of the team.  “I doubt that the seams would have held for more than three or four more hours.”

Kurin inquired, “What about the other four?”

He looked puzzled.  “No need to look further.  We hit it on the first try.”

Softly, Kurin pointed out, “All of the boats in the contest used the same Glue.  Maybe we had better look at the others too.”

It didn’t take genius to see that the seams on all of the classroom boats were starting to open up below the water line.  Kurin scraped out some samples.

Almost without thinking, Kurin descended the ladder to the boat shop.  Greeting the journeyman in charge, she said cheerfully, “Hi, Yoram.  I need the Glue test kit for a few minutes.  Found an anomaly in some glue.  I need to check it out.”  She got out the kit and was starting to mix a test reagent, back turned to him.

Lissa screamed.  Kurin dropped at the warning.  The heavy mallet in Yoram’s big fist clipped the top of her head lightly as it passed through where she had been standing. Kurin uncoiled from the deck, extended left hand chopping at the inside of Yoram’s elbow, the right striking him solidly in the solar plexus.  The mallet went flying and Yoram fell to the deck so hard that the noise brought people from the mess to investigate. Kurin tottered and fell to the floor on hands and knees, slightly dazed from the mallet blow.

The crowd found Kurin just trying to get to her feet.  Yoram still lay on the deck, gasping for breath. Lissa was standing over him with a mallet in her hand.

As men rushed to disarm Lissa, Kurin called out, “Stop!  Yoram attacked me when I came to test this glue from the contest boats.  We will need to convene a Council Trial.”  She showed the crowd her bloodied head.  “If I hadn’t ducked when Lissa screamed, Yoram would have killed me.

“We need someone else, independent of the boat shop to conduct these tests.  I’ve just become part of the proceedings.”

Lissa helped her daughter to her feet.  Still supporting Kurin, Lissa asked, “How did you do that to Yoram?  You are having trouble standing up.”

A still shaking Kurin replied seriously, “Reflex.  Arrakan Jitsu often happens too fast for any thought beyond deciding whether to defend, attack or whatever.  All of that training that Captain Tanlin has been giving me paid off, I guess.”

The trial was held the next day. Yoram and Juris sat facing the whole ship’s company on the main deck, just forward of the tiller walk.  Captain Mord was presiding and Alor had the ship’s case.  The jury was composed of the officers and Masters.

Alor stood and looked sadly at the defendants.  She began simply, “We all know of the many things that Juris has done.  He is a remarkable craftsman.  He has been an excellent leader of our Council.”  She shook her head.  “Every one of us wishes that this was all that need be said.”

TO BE CONTINUED

<==PREVIOUS ~ NEXT==>

[wisdomrays]

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 2&#960; 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.