Question

select the correct answer. what is the corresponding point on the unit circle for the given radian measure? $\theta=\frac{7pi}{6}$ a. $(\frac{1}{2},\frac{sqrt{3}}{2})$ b. $(-\frac{sqrt{3}}{2},-\frac{1}{2})$ c. $(-\frac{1}{2},-\frac{sqrt{3}}{2})$ d. $(\frac{sqrt{3}}{2},\frac{1}{2})$

Explanation:

Step1: Recall unit - circle coordinates formula

For a point \((x,y)\) on the unit circle corresponding to an angle \(\theta\), \(x = \cos\theta\) and \(y=\sin\theta\). Here \(\theta=\frac{7\pi}{6}\).

Step2: Rewrite the angle

We can write \(\frac{7\pi}{6}=\pi+\frac{\pi}{6}\).

Step3: Use trigonometric identities

We know that \(\cos(A + B)=\cos A\cos B-\sin A\sin B\) and \(\sin(A + B)=\sin A\cos B+\cos A\sin B\). For \(A = \pi\) and \(B=\frac{\pi}{6}\), \(\cos(\pi+\frac{\pi}{6})=-\cos\frac{\pi}{6}=-\frac{\sqrt{3}}{2}\) and \(\sin(\pi+\frac{\pi}{6})=-\sin\frac{\pi}{6}=-\frac{1}{2}\).

Answer:

B. \((-\frac{\sqrt{3}}{2},-\frac{1}{2})\)