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Simplifying Trig Expressions using Double and Half Angle Formulas Lori Jordan Kate Dirga Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access a customizable version of this book, as well as other interactive content, visit www.ck12.org CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-source, collaborative, and web-based compilation model, CK-12 pioneers and promotes the creation and distribution of high-quality, adaptive online textbooks that can be mixed, modified and printed (i.e., the FlexBook® textbooks). Copyright © 2016 CK-12 Foundation, www.ck12.org The names “CK-12” and “CK12” and associated logos and the terms “FlexBook®” and “FlexBook Platform®” (collectively “CK-12 Marks”) are trademarks and service marks of CK-12 Foundation and are protected by federal, state, and international laws. Any form of reproduction of this book in any format or medium, in whole or in sections must include the referral attribution link http://www.ck12.org/saythanks (placed in a visible location) in addition to the following terms. Except as otherwise noted, all CK-12 Content (including CK-12 Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution-Non-Commercial 3.0 Unported (CC BY-NC 3.0) License (http://creativecommons.org/ licenses/by-nc/3.0/), as amended and updated by Creative Commons from time to time (the “CC License”), which is incorporated herein by this reference. Complete terms can be found at http://www.ck12.org/about/ terms-of-use. Printed: June 21, 2016 AUTHORS Lori Jordan Kate Dirga www.ck12.org C HAPTER Chapter 1. Simplifying Trig Expressions using Double and Half Angle Formulas 1 Simplifying Trig Expressions using Double and Half Angle Formulas Here you’ll use the half and double angle formulas to simplify more complicated expressions. As Agent Trigonometry, you are given the following cryptic clue. How could you simplify this clue? tan 2x tanx 1+tan x Simplifying Trigonometric Expressions We can also use the double-angle and half-angle formulas to simplify trigonometric expressions. Simplify using the double angle and half angle formulas Simplify cos 2x sin x cos x . Use cos 2a = cos2 a − sin2 a and then factor. cos 2x cos2 x − sin2 x = sin x cos x sin x + cos x ( (( (sin (cos x+ x) (cos x − sin x)( (( = ( ( sin(x( +( cos x ( = cos x − sin x Find the formula for sin 3x. You will need to use the sum formula and the double-angle formula. sin 3x = sin(2x + x) sin 3x = sin(2x + x) = sin 2x cos x + cos 2x sin x = 2 sin x cos x cos x + sin x(2 cos2 x − 1) = 2 sin x cos2 x + 2 sin x cos2 x − sin x = 4 sin x cos2 x − sin x = sin x(4 cos2 x − 1) We will explore other possibilities for the sin 3x because of the different formulas for cos 2a in the Problem Set. Verify the identity cos x + 2 sin2 2x = 1. 1 www.ck12.org Simplify the left-hand side use the half-angle formula. x cos x + 2 sin2 2 !2 r 1 − cos x cos x + 2 2 1 − cos x 2 cos x + 1 − cos x cos x + 2 · 1 Examples Example 1 Earlier, you were asked how could you simplify Use tan 2a = 2 tan a 1−tan2 a tan 2x tanx 1+tan x and then factor. 2 tan x 1 + tan x · 1 − tan2 x tanx 2 tan x 1 + tan x 2 = · = (1 + tan x)(1 − tan x) tanx 1 − tan x tan 2x tanx 1+tan x = Example 2 Simplify sin 2x sin x = sin 2x sin x . 2 sin x cos x sin x = 2 cos x Example 3 Verify cos x + 2 cos2 2x = 1 + 2 cos x. x cos x + 2 cos2 = 1 + 2 cos x 2 r 2 1 + cos x cos x + 2 = 2 cos x + 1 + cos x = 1 + 2 cos x = Review Simplify the following expressions. 1. 2 √ 2 + 2 cos x cos 2x www.ck12.org Chapter 1. Simplifying Trig Expressions using Double and Half Angle Formulas 2x 2. cos cos2 x 3. tan 2x(1 + tan x) 4. cos 2x − 3 sin2 x 2x 5. 1+cos cot x 6. (1 + cos x)2 tan 2x Verify the following identities. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. sin x cot 2x = 1−cos x sin x 1−cos x = 1+cos x sin x sin 2x 1+cos 2x = tan x (sin x + cos x)2 = 1 + sin 2x sin x tan 2x + 2 cos x = 2 cos2 2x cot x + tan x = 2 csc 2x cos 3x = 4 cos3 x − 3 cos x cos 3x = cos3 x − 3 sin2 x cos x sin 2x − tan x = tan x cos 2x cos4 x − sin4 x = cos 2x Answers for Review Problems To see the Review answers, open this PDF file and look for section 14.16. F 3