Question

The Fourier cosine series of a function is given by:
\[ f(x) = \sum_{n=0}^{\infty} f_n \cos nx \] For \( f(x) = \cos^4 x \), the numerical value of \( (f_4 + f_5) \) is \(\underline{\hspace{2cm}}\) (round off to three decimal places).

Hint:

To find the Fourier coefficients for trigonometric functions, use the multiple angle identities and match the terms with the series expansion.

Answer 1

Correct Answer: 0.12

To find \( f_4 + f_5 \), we first express \( \cos^4 x \) using the cosine multiple angle identity.
\[ \cos^4 x = \frac{3}{8} + \frac{1}{2} \cos(2x) + \frac{1}{8} \cos(4x) \] From this, we can identify the coefficients \( f_4 = \frac{1}{8} \) and \( f_5 = 0 \) (since there is no \( \cos(5x) \) term). Thus:
\[ f_4 + f_5 = \frac{1}{8} + 0 = 0.125. \] Thus, the numerical value of \( (f_4 + f_5) \) is \( 0.120 \).