Trigonometric Expansions

The questions below are due on Thursday February 20, 2025; 02:00:00 PM.
 
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In this problem, we compare two methods for expanding a function f(\theta) as a series of the form

f(\theta)=c_0+\sum_{k=1}^\infty c_k\cos(k\theta)+\sum_{k=1}^\infty d_k\sin(k\theta)\,.

Part a. Use the trigonometric identities provided below plus the rules of ordinary algebra to determine the values of the non-zero coefficients c_k and d_k needed to expand the function

f_1(\theta)=\cos^5(\theta)\,.

Part b. An alternative to trigonometric identities is to use complex exponentials. Determine the non-zero coefficients c_k and d_k as in the previous part -- but this time use Euler's formula and complex numbers, but no trigonometric identifies.

Part c. Use the trigonometric identities provided below plus the rules of ordinary algebra to determine the values of the non-zero coeeficients c_k and d_k needed to expand the function

f_2(\theta)=\sin^5(\theta)\,.

Part d. Determine the non-zero coefficients c_k and d_k as in the previous part -- but this time use Euler's formula and complex numbers, but no trigonometric identities.

Part e. List the mathematical relations that you used in each of the previous parts. Briefly describe the pros and cons of using trigonometric identities versus Euler's formula.


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sin(a+b) = sin(a) cos(b) + cos(a) sin(b)
sin(a-b) = sin(a) cos(b) - cos(a) sin(b)
cos(a+b) = cos(a) cos(b) - sin(a) sin(b)
cos(a-b) = cos(a) cos(b) + sin(a) sin(b)
tan(a+b) = (tan(a)+tan(b))/(1-tan(a) tan(b))
tan(a-b) = (tan(a)-tan(b))/(1+tan(a) tan(b))

sin(A) + sin(B) =  2 sin((A+B)/2) cos((A-B)/2)
sin(A) - sin(B) =  2 cos((A+B)/2) sin((A-B)/2)
cos(A) + cos(B) =  2 cos((A+B)/2) cos((A-B)/2)
cos(A) - cos(B) = -2 sin((A+B)/2) sin((A-B)/2)

sin(a+b) + sin(a-b) =  2 sin(a) cos(b)
sin(a+b) - sin(a-b) =  2 cos(a) sin(b)
cos(a+b) + cos(a-b) =  2 cos(a) cos(b)
cos(a+b) - cos(a-b) = -2 sin(a) sin(b)

2 cos(A) cos(B) = cos(A-B) + cos(A+B)
2 sin(A) sin(B) = cos(A-B) - cos(A+B)
2 sin(A) cos(B) = sin(A+B) + sin(A-B)
2 cos(A) sin(B) = sin(A+B) - sin(A-B)

atan2(y1,x1) +/- atan2(y2,x2) = atan2(y1x2 +/- y2x1,x1x2 -/+ y1y2)