Question

select all the correct answers. if the measure of angle $\theta$ is $\frac{11pi}{6}$, which statements are true? $square \tan(\theta)=1$ $square sin(\theta)=\frac{1}{2}$ $square cos(\theta)=\frac{sqrt{3}}{2}$ $square$ the measure of the reference angle is $45^{circ}$. $square$ the measure of the reference angle is $60^{circ}$. $square$ the measure of the reference angle is $30^{circ}$.

Explanation:

Step1: Find the reference - angle

The angle $\theta=\frac{11\pi}{6}$. Since $\frac{11\pi}{6}=2\pi-\frac{\pi}{6}$, the reference - angle $\theta_{r}=\frac{\pi}{6}=30^{\circ}$.

Step2: Calculate trigonometric functions

We know that $\sin(\frac{11\pi}{6})=\sin(2\pi - \frac{\pi}{6})=-\sin(\frac{\pi}{6})=-\frac{1}{2}$, $\cos(\frac{11\pi}{6})=\cos(2\pi - \frac{\pi}{6})=\cos(\frac{\pi}{6})=\frac{\sqrt{3}}{2}$, and $\tan(\frac{11\pi}{6})=\frac{\sin(\frac{11\pi}{6})}{\cos(\frac{11\pi}{6})}=\frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}}=-\frac{1}{\sqrt{3}}$.

Answer:

$\cos(\theta)=\frac{\sqrt{3}}{2}$, The measure of the reference angle is $30^{\circ}$