Question

use the unit - circle, along with the definitions of the circular functions, to find the exact values for the given functions when s=-2π. sin(-2π), cos(-2π), tan(-2π). select the correct choice below and fill in any answer boxes in your choice. a. sin(-2π)=0 (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.) b. the solution is undefined. select the correct choice below and fill in any answer boxes in your choice. a. cos(-2π)= (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.) b. the solution is undefined.

Explanation:

Step1: Recall cosine - angle property

Cosine is an even function, i.e., $\cos(-\theta)=\cos(\theta)$. So, $\cos(- 2\pi)=\cos(2\pi)$.

Step2: Use unit - circle definition

On the unit - circle, for an angle $\theta = 2\pi$, the $x$ - coordinate (which is the value of $\cos\theta$) is 1. So, $\cos(2\pi)=1$.

Step3: Recall sine - angle property

Sine is an odd function, i.e., $\sin(-\theta)=-\sin(\theta)$. So, $\sin(-2\pi)=-\sin(2\pi)$.

Step4: Use unit - circle definition for sine

On the unit - circle, for an angle $\theta = 2\pi$, the $y$ - coordinate (which is the value of $\sin\theta$) is 0. So, $\sin(2\pi) = 0$ and $\sin(-2\pi)=0$.

Step5: Calculate tangent

The tangent function is defined as $\tan\theta=\frac{\sin\theta}{\cos\theta}$. For $\theta=-2\pi$, $\tan(-2\pi)=\frac{\sin(-2\pi)}{\cos(-2\pi)}$. Since $\sin(-2\pi) = 0$ and $\cos(-2\pi)=1$, then $\tan(-2\pi)=0$.

Answer:

$\sin(-2\pi)=0$, $\cos(-2\pi)=1$, $\tan(-2\pi)=0$