Final answer:
To convert x(t) = sin³(27πt) to a sum of complex exponentials, apply a trigonometric identity to express sin³(θ) in terms of sin(θ) and sin(3θ), then use Euler's formula to represent these terms as complex exponentials.
Step-by-step explanation:
The student's question involves finding a formula for the function x(t) = sin3(27πt) as the real part of a sum of complex exponentials. To solve this, one can use trigonometric identities to express sin3(θ) in terms of multiple angles and then represent those using Euler's formula for complex exponentials. For instance, the identity sin3(θ) = (3sin(θ) - sin(3θ))/4 can be combined with Euler's formula eiθ = cos(θ) + isin(θ) to express sin3(27πt) in terms of exponentials. Then, the desired formula can be written as a sum of terms involving eiαt and their conjugates, where the real part of this sum yields the original function x(t).