Question

Given f(x) = sin x, evaluate the function at x = 13pi/2

Answer 1

To evaluate \( f(x) = \sin x \) at \( x = \frac{13\pi}{2} \), we can first simplify \( \frac{13\pi}{2} \) by reducing it to an equivalent angle within the principal range of the sine function, which is \( [0, 2\pi) \).

1. Find the equivalent angle by subtracting multiples of \( 2\pi \) from \( \frac{13\pi}{2} \):
\[
2\pi = \frac{4\pi}{1} \equiv \frac{8\pi}{2}
\]
So, subtract \( 2\pi \) twice to see where we land:
\[
\frac{13\pi}{2} - 2(2\pi) = \frac{13\pi}{2} - \frac{8\pi}{2} = \frac{5\pi}{2}
\]

2. Now, we need to reduce \( \frac{5\pi}{2} \):
\[
\frac{5\pi}{2} - 2\pi = \frac{5\pi}{2} - \frac{4\pi}{2} = \frac{\pi}{2}
\]

3. Now we can evaluate \( \sin \left( \frac{5\pi}{2} \right) \) which is equivalent to \( \sin \left( \frac{\pi}{2} \right) \):
\[
\sin \left( \frac{\pi}{2} \right) = 1
\]

Thus, the final result is:
\[
f\left( \frac{13\pi}{2} \right) = \sin \left( \frac{13\pi}{2} \right) = 1
\]