Question

use reference angles to find the exact value of the following expression. cos(11π/6). determine the reference angle for 11π/6. the reference angle is _. (type your answer in radians. use integers or fractions for any numbers in the expression. type an exact answer, using π as needed.)

Explanation:

Step1: Determine the quadrant of the angle

The angle $\theta=\frac{11\pi}{6}$ is in the fourth - quadrant since $2\pi-\frac{\pi}{6}=\frac{12\pi - \pi}{6}=\frac{11\pi}{6}$, and angles in the range $\frac{3\pi}{2}<\theta < 2\pi$ are in the fourth - quadrant.

Step2: Calculate the reference angle

The reference angle $\theta_{r}$ for an angle $\theta$ in the fourth - quadrant is given by $2\pi-\theta$. For $\theta = \frac{11\pi}{6}$, the reference angle $\theta_{r}=2\pi-\frac{11\pi}{6}=\frac{\pi}{6}$.

Step3: Find the cosine value

We know that $\cos\theta$ has the same absolute value as $\cos\theta_{r}$ in the fourth - quadrant, and $\cos$ is positive in the fourth - quadrant. Since $\cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}$, then $\cos\frac{11\pi}{6}=\frac{\sqrt{3}}{2}$.

Answer:

The reference angle is $\frac{\pi}{6}$ and $\cos\frac{11\pi}{6}=\frac{\sqrt{3}}{2}$