Explanation Given: cos π 8 = 1 2 2 + 2 The given equation can be proved by using the double-angle formula Basic trigonometric identities : Pythagorean identity sin 2 x + cos 2 x = 1 Complementary angles 1. sin ( π 2 − x ) = cos x 2. cos ( π 2 − x ) = sin x Addition formulas 1. sin ( x + y ) = sin x cos y + cos x sin y 2. cos ( x + y ) = cos x cos y − sin x sin y Double-angle formulas 1. sin 2 x = 1 2 ( 1 − cos 2 x ) 2. cos 2 x = 1 2 ( 1 + cos 2 x ) 3. cos 2 x = cos 2 x − sin 2 x 4. sin 2 x = 2 sin x cos x Shift formulas 1. sin ( x + π 2 ) = cos x 2. cos ( x + π 2 ) = − sin x Reference Trigonometry table of values Radians 0 π 6 π 4 π 3 π 2 sin θ 0 1 2 1 2 3 2 1 cos θ 1 3 2 1 2 1 2 0 tan θ 0 1 3 1 3 ∞ To prove the given equation cos π 8 = 1 2 2 + 2 , let us use the double angle formula 2 To determine (b) The proof of the given equation by applying the double-angle formula and guess the values of cos π 32 and cos π 2 n for all n.

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Fundamentals of Trigonometric Identities

Trigonometric Identities are the equalities that involve trigonometry functions and hold for all the values of variables given in the equation.

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Question

Chapter 1.4, Problem 66E

To determine

(a)

The proof of the given equation by applying the double-angle formula

To determine

(b)

The proof of the given equation by applying the double-angle formula and guess the values of cosπ32 and cosπ2n for all n.

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Chapter 1.4, Problem 65E

Chapter 1.5, Problem 1PQ

Chapter 1 Solutions

Calculus - Standalone book

Ch. 1.1 - Prob. 1PQ Ch. 1.1 - Prob. 2PQ Ch. 1.1 - Prob. 3PQ Ch. 1.1 - Prob. 4PQ Ch. 1.1 - Prob. 5PQ Ch. 1.1 - Prob. 6PQ Ch. 1.1 - Prob. 7PQ Ch. 1.1 - Prob. 8PQ Ch. 1.1 - Prob. 9PQ Ch. 1.1 - Prob. 10PQ

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(a) The proof of the given equation by applying the double-angle formula

Solution Summary: The author explains how the given equation can be proved by using the double-angle formula.