Exploring Trigonometric Identities

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Questions and Answers

किस संख्या के लिए, $ heta = 30^ ext{o} $ और $ x = rac{ oot3 ext{ }}{2} $ सही है?

$ heta = 45^ ext{o} $ और $ x = rac{1}{2} $ (correct)
$ heta = 60^ ext{o} $ और $ x = rac{1}{2} $
$ heta = 60^ ext{o} $ और $ x = rac{ oot3 ext{ }}{2} $
$ heta = 45^ ext{o} $ और $ x = rac{ oot3 ext{ }}{2} $

कौन सी पहचान एकत्रित कोणों के समीकरण को प्रस्तुत करती है?

$ an(x + y) = \frac{\tan(x) + \tan(y)}{1 - \tan(x)\tan(y)} $
$ an(x + y) = \frac{\tan(x) + \tan(y)}{1 + \tan(x)\tan(y)} $ (correct)
$ an(x - y) = \frac{\tan(x) - \tan(y)}{1 + \tan(x)\tan(y)} $
$ an(x - y) = \frac{\tan(x) - \tan(y)}{1 - \tan(x)\tan(y)} $

कौन सा उल्टावलाँबांतर व्यक्ति का समीकरण सही है?

$ \dfrac{1}{\tan^2(x)} = 1 - \cot^2(x) $
$ an^2(x) = \cot^2(x) - 1 $
$ an^2(x) = 1 + \cot^2(x) $
$ an^2(x) = \dfrac{1}{1 + \cot^2(x)} $ (correct)

क्या है डबल-कोण समीकरण, $ an(2x) $ का सही समीकरण?

$ 2\tan(x) \left(1 - \tan^2(x)\right) $ (C) Signup and view all the answers

किस समीकरण में दिए गए, 'पूरक' कोणों का समीकरण सही है?

$ an(90 - x) = \cot(x) $ (B) Signup and view all the answers

किस 'पूरक-समीकरण' में, 'समान-पुन:प्राप्त' के संस्करण को प्रस्तुत किया जा सकता है?

$ 2 heta = x - y, 2x = 90 - y, 2y = 90 + x $ (D) Signup and view all the answers

किस त्रिकोणमिति समीकरण को प्यथागोरियन त्रिकोणमिति कहा जाता है?

( \sin^2(x) + \cos^2(x) = 1 ) (A) Signup and view all the answers

त्रिकोणमिति समीकरण क्या होते हैं?

त्रिकोणमिति समीकरण वह समीकरण हैं जो समान कोण के विभिन्न त्रिकोणमिति कार्यों को संबंधित करते हैं। (C) Signup and view all the answers

कौन सी त्रिकोणमिति व्यासन त्रिकोणमिति होती है?

( \frac{1}{\cos(x)} = \sec(x) ) (A) Signup and view all the answers

त्रिकोणमिति में परिस्थिति त्रिकोणमिति कहलाते है?

( \cos(-x) = \cos(x) ) (C) Signup and view all the answers

त्रिकोणमिति में चिह्न परिवर्तन की त्रिकोणमिति क्या है?

( \sin(\pi - x) = -\sin(x) ) (A) Signup and view all the answers

त्रिकोणमिति में 'संकलन' के लिए प्रमुख व्यासन है:

( \frac{1}{\cos(x)} = \sec(x) ) (D) Signup and view all the answers

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Study Notes

Exploring Trigonometric Identities

Trigonometry, a branch of mathematics that deals with angles and their relationships to the lengths and ratios of the sides of triangles, also encompasses a set of powerful tools known as trigonometric identities. These identities help us simplify and manipulate trigonometric expressions and equations, enabling us to solve problems that would be otherwise impossible.

What are Trigonometric Identities?

Trigonometric identities are equations that relate the various trigonometric functions of the same angle, or different angles. They can be used to find new expressions for trigonometric functions, replace one trigonometric function with another, or to generate different forms of a given trigonometric expression.

Basic Trigonometric Identities

Reciprocal Identities [ \frac{1}{\sin(x)} = \csc(x), \quad \frac{1}{\cos(x)} = \sec(x), \quad \frac{1}{\tan(x)} = \cot(x) ]
Pythagorean Identity [ \sin^2(x) + \cos^2(x) = 1 ]
Symmetry Identities [ \sin(-x) = -\sin(x), \quad \cos(-x) = \cos(x), \quad \tan(-x) = -\tan(x) ]
Sign Change Identities [ \sin(\pi - x) = \sin(x), \quad \cos(\pi - x) = -\cos(x), \quad \tan(\pi - x) = -\tan(x) ]

Important Trigonometric Identities

Product-to-Sum Identities [ \sin(x + y) = \sin(x) \cos(y) + \cos(x) \sin(y), \quad \sin(x - y) = \sin(x) \cos(y) - \cos(x) \sin(y) ] [ \cos(x + y) = \cos(x) \cos(y) - \sin(x) \sin(y), \quad \cos(x - y) = \cos(x) \cos(y) + \sin(x) \sin(y) ]
Double-Angle Identities [ \sin(2x) = 2 \sin(x) \cos(x), \quad \cos(2x) = \cos^2(x) - \sin^2(x) ]
Sum-to-Product Identities [ \sin(x + y) = \sin(x) \cos(y) + \cos(x) \sin(y), \quad \sin(x - y) = \sin(x) \cos(y) - \cos(x) \sin(y) ] [ \cos(x + y) = \cos(x) \cos(y) - \sin(x) \sin(y), \quad \cos(x - y) = \cos(x) \cos(y) + \sin(x) \sin(y) ]
Trigonometric Substitution [ \sqrt{1 - x^2} = \cos(\theta), \quad x = \sin(\theta) ]
Reduction Formulae [ \sin(n x) = 2 \sin(x) \cos((n - 1) x) - \sin((n - 2) x), \quad n \geq 2 ] [ \cos(n x) = \cos(2 x) \cos((n - 1) x) - \sin(2 x) \sin((n - 1) x), \quad n \geq 2 ]

Uses of Trigonometric Identities

Trigonometric identities are a powerful tool in solving a wide variety of problems. They can be used to rewrite expressions in terms of other functions, find roots or zeros of trigonometric functions, or to manipulate formulas to simplify calculations.

Summary

The study of trigonometric identities enriches our understanding of trigonometry and provides us with an arsenal of tools to solve complex problems. While the list of identities presented here is not exhaustive, it includes many of the most commonly used identities in trigonometry. Mastering these identities will provide you with the ability to tackle more advanced problems and expand your knowledge of this fascinating field.