We are tasked with finding the exact value of the composite function $g^{-1}(f(-\frac{\pi}{4}))$.

Step 1: Evaluate $f(-\frac{\pi}{4})$

We are given the function: $f(x) = \sin(x)$ Now, substitute $x = -\frac{\pi}{4}$ into the sine function: $f\left(-\frac{\pi}{4}\right) = \sin\left(-\frac{\pi}{4}\right)$ Using the identity $\sin(-\theta) = -\sin(\theta)$, we know that: $\sin\left(-\frac{\pi}{4}\right) = -\sin\left(\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}$ Thus: $f\left(-\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}$

Step 2: Solve for $g^{-1}(f(-\frac{\pi}{4})) = g^{-1}\left(-\frac{\sqrt{2}}{2}\right)$

We are given the function: $g(x) = \cos(x)$ Now, we need to find the inverse function, $g^{-1}(y)$, such that $g(g^{-1}(y)) = y$. This means we are looking for $x$ such that: $\cos(x) = -\frac{\sqrt{2}}{2}$ We know that $\cos(x) = -\frac{\sqrt{2}}{2}$ at $x = \frac{3\pi}{4}$, within the interval $0 \leq x \leq \pi$.

Thus: $g^{-1}\left(-\frac{\sqrt{2}}{2}\right) = \frac{3\pi}{4}$

Final Answer:

The exact value of $g^{-1}(f(-\frac{\pi}{4}))$ is: $\frac{3\pi}{4}$

Would you like more details on any steps?

Here are five related questions:

1. How do we determine the inverse of a trigonometric function?
2. What is the domain and range of $\cos(x)$ within $0 \leq x \leq \pi$?
3. How do we find the exact value of trigonometric functions at specific angles?
4. What is the significance of the interval restrictions for $x$ in trigonometric functions?
5. How do we use trigonometric identities to simplify expressions?

Tip: Always check the domain of the function and its inverse to ensure you're using the correct range for inverse trigonometric values.