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identities: 1. sin(3x) = (sin(x))(4(cos(x))² – 1)

identities: 1. sin(3x) = (sin(x))(4(cos(x))² – 1)

Algebra and Trigonometry (MindTap Course List)
Algebra and Trigonometry (MindTap Course List)
4th Edition
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter7: Analytic Trigonometry
Section7.4: Basic Trigonometric Equations
Problem 2E: The basic equation sinx=2 has_________no/one/infinitely many solutions, whereas the basic equation...
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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))? – 1 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 = + 2 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)
Transcribed Image Text: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))? – 1 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 = + 2 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)
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