Question

end of semester test
6
select all the correct answers.
if the measure of angle is $\theta$ is $\frac{7pi}{4}$, which statements are true?
$square \tan(\theta)= - 1$
$square$ the measure of the reference angle is $45^{circ}$.
$square cos(\theta)=-\frac{sqrt{2}}{2}$
$square$ the measure of the reference angle is $30^{circ}$.
$square$ the measure of the reference angle is $60^{circ}$.
$square sin(\theta)=-\frac{sqrt{2}}{2}$

Explanation:

Step1: Determine the quadrant of the angle

The angle $\theta=\frac{7\pi}{4}= 315^{\circ}$, which lies in the fourth - quadrant.

Step2: Find the reference angle

The reference angle $\theta_{r}$ for an angle $\theta$ in the fourth - quadrant is given by $2\pi-\theta$ (in radians) or $360^{\circ}-\theta$ (in degrees). So, $360 - 315=45^{\circ}$ or $\frac{8\pi}{4}-\frac{7\pi}{4}=\frac{\pi}{4}$ radians.

Step3: Calculate trigonometric functions

We know that $\sin(\frac{7\pi}{4})=-\frac{\sqrt{2}}{2}$, $\cos(\frac{7\pi}{4})=\frac{\sqrt{2}}{2}$, and $\tan(\frac{7\pi}{4})=\frac{\sin(\frac{7\pi}{4})}{\cos(\frac{7\pi}{4})}=\frac{-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}=-1$.

Answer:

• $\tan(\theta)= - 1$
• The measure of the reference angle is $45^{\circ}$
• $\sin(\theta)=-\frac{\sqrt{2}}{2}$