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2. cos(4x) = 1 – 8(sin(x))²(cos(x))²

2. cos(4x) = 1 – 8(sin(x))²(cos(x))²

Trigonometry (11th Edition)
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE: 1. Give the measures of the complement and the supplement of an angle measuring 35°.
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Double and Half Angles Let's start with the formula for the cosine of a sum: cos(A + B) = cos(A) cos(B) – sin(A) sin(B) Now replace B with A and simplify and we get a double angle formula: (1) cos(2A) = (cos(A))² – (sin(A))² (2) %3D Now do the same with sine: sin(A + B) = sin(A) cos(B) + cos(A) sin(B) (3) So, sin(2A) = 2 sin(A) cos(B) (4) To get half angle formulas we start with (2) above and replace (sin(A))² with 1- (cos(A))? to get cos(2A) = 2(cos(A))² – 1 When we solve this equation for cos(A), we get 1+cos(2A) cos(A) = ±, 2 Finally, replace A with to get the half angle formula: 2 |1+cos(0) cos = + (5) In this formula, you have consider the quadrant of to decide if you should keep the plus or minus sign. See if you can start with (2) above and derive the other half angle formula: 1-cos(0) sin ) = +, 2 Now see if you can use your higher order thinking skills to prove the following identities:
Transcribed Image Text:Double and Half Angles Let's start with the formula for the cosine of a sum: cos(A + B) = cos(A) cos(B) – sin(A) sin(B) Now replace B with A and simplify and we get a double angle formula: (1) cos(2A) = (cos(A))² – (sin(A))² (2) %3D Now do the same with sine: sin(A + B) = sin(A) cos(B) + cos(A) sin(B) (3) So, sin(2A) = 2 sin(A) cos(B) (4) To get half angle formulas we start with (2) above and replace (sin(A))² with 1- (cos(A))? to get cos(2A) = 2(cos(A))² – 1 When we solve this equation for cos(A), we get 1+cos(2A) cos(A) = ±, 2 Finally, replace A with to get the half angle formula: 2 |1+cos(0) cos = + (5) In this formula, you have consider the quadrant of to decide if you should keep the plus or minus sign. See if you can start with (2) above and derive the other half angle formula: 1-cos(0) sin ) = +, 2 Now see if you can use your higher order thinking skills to prove the following identities:
2. cos(4x) = 1 – 8(sin(x))²(cos(x))²
Transcribed Image Text:2. cos(4x) = 1 – 8(sin(x))²(cos(x))²
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