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3. sin(3x) = (sin(x))(3 – 4(sin(x))²) %3D

3. sin(3x) = (sin(x))(3 – 4(sin(x))²) %3D

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) 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 – When we solve this equation for cos(A), we get |1+cos(2A) cos(A) = Finally, replace A with - to get the half angle formula: |1+cos(0) Cos = + (5) 2 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 Now see if you can use your higher order thinking skills to prove the following identities: 1. sin(3x) = (sin(x))(4(cos(x))² – 1) 2. cos(4x) = 1– 8(sin(x))²(cos(x))² 3. sin(3x) = (sin(x))(3 – 4(sin(x))²) 4. sin(4x) = (4 sin(x) cos(x))(2(cos(x))² – 1) Finally, find the following without a calculator: 5. tan 6. cos()
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) 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 – When we solve this equation for cos(A), we get |1+cos(2A) cos(A) = Finally, replace A with - to get the half angle formula: |1+cos(0) Cos = + (5) 2 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 Now see if you can use your higher order thinking skills to prove the following identities: 1. sin(3x) = (sin(x))(4(cos(x))² – 1) 2. cos(4x) = 1– 8(sin(x))²(cos(x))² 3. sin(3x) = (sin(x))(3 – 4(sin(x))²) 4. sin(4x) = (4 sin(x) cos(x))(2(cos(x))² – 1) Finally, find the following without a calculator: 5. tan 6. cos()
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