Trigonometric Series for Trigonometric Functions

The questions below are due on Monday March 01, 2021; 10:00:00 PM.

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Note that this exercise is intended to be solved by hand, without the use of computation.

Determine Fourier series of the form

f(t)=c_0+\sum_{k=1}^\infty c_k\cos\left({2\pi k\over T}t\right)+\sum_{k=1}^\infty d_k\sin\left({2\pi k\over T} t\right)
for the following functions.

\begin{aligned} f_1(t) &= \cos^5(t)\\ f_2(t) &= \sin^5(t)\\ f_3(t) &= \sin^3(t)\cos^2(t)\\ f_4(t) &= \sin^2(t)\cos^3(t) \end{aligned}

Enter a python expression for the c_0 term of f_1. The constant \pi has been imported as pi.
Enter a python list of the c_k terms of f_1 for k\geq1. You may omit trailing zeroes.
(For example, if c_1 = 0, c_2 = \sqrt{2}, c_3 = 3/2, c_4 = 0, c_5 = 1, c_6 = 0, c_7 = 0, c_8 = 0, you should enter [0, 2**0.5, 3/2, 0, 1].)
Enter a python list of the d_k terms of f_1 for k\geq1. You may omit trailing zeroes.

Enter a python expression for the c_0 term of f_2.
Enter a python list of the c_k terms of f_2 for k\geq1. You may omit trailing zeroes.
Enter a python list of the d_k terms of f_2 for k\geq1. You may omit trailing zeroes.

Enter a python expression for the c_0 term of f_3.
Enter a python list of the c_k terms of f_3 for k\geq1. You may omit trailing zeroes.
Enter a python list of the d_k terms of f_3 for k\geq1. You may omit trailing zeroes.

Enter a python expression for the c_0 term of f_4.
Enter a python list of the c_k terms of f_4 for k\geq1. You may omit trailing zeroes.
Enter a python list of the d_k terms of f_4 for k\geq1. You may omit trailing zeroes.