Question

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

Explanation:

Step1: Convert angle to degrees

We know that $\theta=\frac{2\pi}{3}$ radians. Since $1$ radian $=\frac{180^{\circ}}{\pi}$, then $\theta=\frac{2\pi}{3}\times\frac{180^{\circ}}{\pi}=120^{\circ}$.

Step2: Find the reference - angle

The angle $\theta = 120^{\circ}$ is in the second quadrant. The reference - angle $\theta_{r}=180^{\circ}-120^{\circ}=60^{\circ}$.

Step3: Calculate trigonometric values

For $\theta = 120^{\circ}$ (or $\frac{2\pi}{3}$ radians), $\cos(\theta)=\cos(120^{\circ})=-\frac{1}{2}$, $\sin(\theta)=\sin(120^{\circ})=\frac{\sqrt{3}}{2}$, and $\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}=\frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}}=-\sqrt{3}$.

Answer:

The measure of the reference angle is $60^{\circ}$, $\tan(\theta)=-\sqrt{3}$