Question

use identities to write each expression as a function of $\theta$. $\cos(\theta + \frac{\pi}{2})$ a. $-\sin\theta$ b. $\sin\theta$ c. $\cos\theta$ d. $-\cos\theta$

Explanation:

Step1: Use the cosine addition formula

The cosine addition formula is $\cos(A + B)=\cos A\cos B-\sin A\sin B$. Here $A = \theta$ and $B=\frac{\pi}{2}$.
So, $\cos(\theta+\frac{\pi}{2})=\cos\theta\cos\frac{\pi}{2}-\sin\theta\sin\frac{\pi}{2}$.

Step2: Substitute the values of trigonometric functions

We know that $\cos\frac{\pi}{2} = 0$ and $\sin\frac{\pi}{2}=1$.
Substituting these values, we get $\cos\theta\times0-\sin\theta\times1$.

Step3: Simplify the expression

$\cos\theta\times0-\sin\theta\times1=0 - \sin\theta=-\sin\theta$.

Answer:

A. $-\sin\theta$