SOLUTION: Find the sum 𝑆 = cos(𝑥 + 𝜃) + cos(2𝑥 + 3𝜃) + cos(3𝑥 + 5𝜃) + cos(4𝑥 + 7𝜃) + ⋯ + cos(40𝑥 + 79𝜃), and express your answer as a product and quotien

Algebra ->  Trigonometry-basics -> SOLUTION: Find the sum 𝑆 = cos(𝑥 + 𝜃) + cos(2𝑥 + 3𝜃) + cos(3𝑥 + 5𝜃) + cos(4𝑥 + 7𝜃) + ⋯ + cos(40𝑥 + 79𝜃), and express your answer as a product and quotien      Log On


   

Question 1204376: Find the sum
𝑆 = cos(𝑥 + 𝜃) + cos(2𝑥 + 3𝜃) + cos(3𝑥 + 5𝜃) + cos(4𝑥 + 7𝜃) + ⋯ + cos(40𝑥 + 79𝜃),
and express your answer as a product and quotient of trigonometric functions.
Answer by Edwin McCravy(20065) About Me  (Show Source):
You can put this solution on YOUR website!

S%22%22=%22%22

S%22%22=%22%22sum%28%28k%2B1%29x%2B%282k%2B1%29theta%2Ck=0%2C40-1%29

We use this formula from 

https://math.stackexchange.com/questions/17966/how-can-we-sum-up-sin-and-cos-series-when-the-angles-are-in-arithmetic-pro

sum%28cos%28a%2Bkd%29%2Ck=0%2Cn-1%29%22%22=%22%22expr%28sin%28n%2Aexpr%28d%2F2%29%29%2Fsin%28d%2F2%29%29%22%22%2A%22%22cos%28%282a%2B%28n-1%29d%29%2F2%29

We rewrite 
%28k%2B1%29x%2B%282k%2B1%29theta%22%22=%22%22kx%2Bx%2B2k%2Atheta%2Btheta%22%22=%22%22k%2Btheta%2Bkx%2B2k%2Atheta%29%22%22=%22%22%28x%2Btheta%29%2Bk%28x%2B2theta%29

So we substitute a=x%2Btheta, d=x%2B2theta, n=40

sum%28cos%28a%2Bkd%29%2Ck=0%2Cn-1%29%22%22=%22%22expr%28sin%28n%2Aexpr%28d%2F2%29%29%2Fsin%28d%2F2%29%29%22%22%2A%22%22cos%28%282a%2B%28n-1%29d%29%2F2%29


sum%28cos%28%28x%2Btheta%29%2Bk%28x%2B2theta%29%29%2Ck=0%2C40-1%29%22%22=%22%22expr%28sin%2840%2Aexpr%28%28x%2B2theta%29%2F2%29%29%2Fsin%28%28x%2B2theta%29%2F2%29%29%22%22%2A%22%22cos%28%282%28x%2Btheta%29%2B%2840-1%29%28x%2B2theta%29%29%2F2%29%22%22=%22%22expr%28sin%2840%2Aexpr%28%28x%2B2theta%29%2F2%29%29%2Fsin%28%28x%2B2theta%29%2F2%29%29%22%22%2A%22%22cos%28%282x%2B2theta%2B39x%2B78theta%29%2F2%29

S%22%22=%22%22

Edwin