Question

use the unit circle and the period of the function to evaluate the expression. write your answer as a simplified fraction, if necessary. select “undefined” if applicable. part 1 of 3 (a) cos(9π/2)=0 part: 1 / 3 part 2 of 3 (b) sin510°=

Explanation:

Step1: Rewrite angle for cosine

The period of the cosine function is $2\pi$. We can rewrite $\frac{9\pi}{2}$ as $\frac{9\pi}{2}=4\pi+\frac{\pi}{2}$. Since adding a multiple of $2\pi$ to an angle does not change the value of the cosine - function, $\cos(\frac{9\pi}{2})=\cos(4\pi + \frac{\pi}{2})=\cos(\frac{\pi}{2})$. On the unit - circle, at $\theta=\frac{\pi}{2}$, the $x$ - coordinate (which is the value of $\cos\theta$) is 0.

Step2: Rewrite angle for sine

The period of the sine function is $360^{\circ}$. We can rewrite $510^{\circ}$ as $510^{\circ}=360^{\circ}+150^{\circ}$. Since adding a multiple of $360^{\circ}$ to an angle does not change the value of the sine - function, $\sin(510^{\circ})=\sin(360^{\circ}+150^{\circ})=\sin(150^{\circ})$. And $150^{\circ}$ is in the second quadrant, and $\sin(150^{\circ})=\sin(180^{\circ} - 30^{\circ})$. Using the identity $\sin(A - B)=\sin A\cos B-\cos A\sin B$ with $A = 180^{\circ}$ and $B = 30^{\circ}$, or simply knowing the unit - circle values, $\sin(150^{\circ})=\frac{1}{2}$.

Answer:

(a) 0
(b) $\frac{1}{2}$