Question

question given the following unit circle, rotate green dot to the appropriate angle and then find the exact value of the function. \\( \cos 150^{\circ} \\)

Explanation:

Step1: Determine the reference angle

The angle $150^\circ$ is in the second quadrant. The reference angle $\theta_{ref}$ is calculated as $180^\circ - 150^\circ = 30^\circ$.

Step2: Recall the cosine value of the reference angle

We know that $\cos 30^\circ=\frac{\sqrt{3}}{2}$.

Step3: Determine the sign of the cosine in the second quadrant

In the second quadrant, the cosine function is negative (since $x$-coordinates on the unit circle are negative there). So, $\cos 150^\circ = -\cos 30^\circ$.

Step4: Substitute the value of $\cos 30^\circ$

Substituting $\cos 30^\circ=\frac{\sqrt{3}}{2}$ into the equation, we get $\cos 150^\circ = -\frac{\sqrt{3}}{2}$.

Answer:

$\boldsymbol{-\frac{\sqrt{3}}{2}}$