Find the convolution ​f(t)*g(t) for the following problem.
f(t) = g(t) = 7*sin(t)
(f*g)(t)=_________?

To find the convolution \((f * g)(t)\) of the functions \(f(t) = 7 \sin(t)\) and \(g(t) = 7 \sin(t)\), we use the definition of the convolution of two functions:

\[
(f * g)(t) = \int_{-\infty}^{\infty} f(\tau) g(t - \tau) \, d\tau
\]

Given \(f(t) = 7 \sin(t)\) and \(g(t) = 7 \sin(t)\), the convolution integral becomes:

\[
(f * g)(t) = \int_{-\infty}^{\infty} 7 \sin(\tau) \cdot 7 \sin(t - \tau) \, d\tau
\]

We can simplify the integral by factoring out constants:

\[
(f * g)(t) = 49 \int_{-\infty}^{\infty} \sin(\tau) \sin(t - \tau) \, d\tau
\]

To proceed, we use the product-to-sum identities for sine functions. The product-to-sum identities state:

\[
\sin(A) \sin(B) = \frac{1}{2} [\cos(A - B) - \cos(A + B)]
\]

Applying this to our integral, with \(A = \tau\) and \(B = t - \tau\):

\[
\sin(\tau) \sin(t - \tau) = \frac{1}{2} [\cos(\tau - (t - \tau)) - \cos(\tau + (t - \tau))]
\]

\[
= \frac{1}{2} [\cos(2\tau - t) - \cos(t)]
\]

So the integral becomes:

\[
(f * g)(t) = 49 \int_{-\infty}^{\infty} \frac{1}{2} [\cos(2\tau - t) - \cos(t)] \, d\tau
\]

\[
= \frac{49}{2} \left( \int_{-\infty}^{\infty} \cos(2\tau - t) \, d\tau - \int_{-\infty}^{\infty} \cos(t) \, d\tau \right)
\]

Since \(\cos(t)\) is a constant with respect to \(\tau\), the second integral is improper and evaluates to 0 because the integral of a constant over an infinite interval is undefined. Therefore, we focus on the first integral:

\[
\int_{-\infty}^{\infty} \cos(2\tau - t) \, d\tau
\]

This integral is zero because \(\cos(2\tau - t)\) is an oscillatory function with no net area over an infinite interval.

Therefore, the convolution \((f * g)(t)\) evaluates to zero:

\[
(f * g)(t) = 0
\]

So the result is:

\[
(f * g)(t) = 0
\]

Would you like a detailed explanation or have any questions about the steps?

Here are some related questions for further exploration:

1. What is the definition of the convolution of two functions?
2. How do you apply the product-to-sum identities for trigonometric functions?
3. Why does the integral of \(\cos(2\tau - t)\) over an infinite interval evaluate to zero?
4. What are some properties of convolution in the context of signal processing?
5. How does convolution relate to the Fourier transform?
6. What is the significance of the result when the convolution of two functions is zero?
7. How can we interpret the convolution result in terms of signal overlap?
8. What are some common applications of convolution in engineering and physics?

**Tip:** When dealing with convolutions involving sine and cosine functions, using trigonometric identities can greatly simplify the process.

Learn how to calculate the convolution (f * g)(t) for the functions f(t) = 7*sin(t) and g(t) = 7*sin(t) using trigonometric identities and integral calculus. Detailed step-by-step solution provided.

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