Question

Express the function f(t) = sin⁴(t) in its Fourier series in exponential, trigonometric, and compact forms.

a) (1/8)e^(4it) - (1/2)e^(2it) + (3/8) + (1/2)e^(-2it) - (1/8)e^(-4it)
b) (1/2) - (1/2)cos(2t) + (3/8)cos(4t)
c) (1/2)e^(2it) + (1/2)e^(-2it)
d) (1/2)e^(it) - (1/2)e^(-it) + (1/2)e^(3it) - (1/2)e^(-3it)

Answer 1

The function f(t) = sin⁴(t) can be expressed in its Fourier series in exponential, trigonometric, and compact forms using trigonometric identities.

Step-by-step explanation:

To express the function f(t) = sin⁴(t) in its Fourier series in exponential, trigonometric, and compact forms, we can use the following trigonometric identities:

• sin²(x) = (1 - cos(2x)) / 2
• cos(2x) = 2cos²(x) - 1

Applying these identities to the function f(t), we get:

• Exponential form: (1/8)e^(4it) - (1/2)e^(2it) + (3/8) + (1/2)e^(-2it) - (1/8)e^(-4it)
• Trigonometric form: (1/2) - (1/2)cos(2t) + (3/8)cos(4t)
• Compact form: (1/2)e^(2it) + (1/2)e^(-2it)