Use reference angles to find the exact value of the following expression. $$ \cos \frac{7 \pi}{6} $$ Select the correct choice below and, if necessary, fill in the answer box to gomplete your choice. A. $$ \cos \frac{7 \pi}{6}=\square $$ (Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions the denominator.)

Question

UpStudy AI · Accepted Answer

To find the exact value of $$ \cos \frac{7 \pi}{6} $$, we can use the concept of reference angles.

1. **Identify the angle**: The angle $$ \frac{7 \pi}{6} $$ is in the third quadrant because it is greater than $$ \pi $$ (or $$ \frac{6 \pi}{6} $$) and less than $$ \frac{3\pi}{2} $$ (or $$ \frac{9 \pi}{6} $$).

2. **Find the reference angle**: The reference angle for an angle in the third quadrant can be found by subtracting $$ \pi $$ from the angle:
   $$
   \text{Reference angle} = \frac{7 \pi}{6} - \pi = \frac{7 \pi}{6} - \frac{6 \pi}{6} = \frac{\pi}{6}
   $$

3. **Determine the cosine value**: The cosine of an angle in the third quadrant is negative. The cosine of the reference angle $$ \frac{\pi}{6} $$ is:
   $$
   \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}
   $$
   Therefore, since $$ \frac{7 \pi}{6} $$ is in the third quadrant:
   $$
   \cos \frac{7 \pi}{6} = -\cos \frac{\pi}{6} = -\frac{\sqrt{3}}{2}
   $$

Thus, the exact value of $$ \cos \frac{7 \pi}{6} $$ is:
$$
\cos \frac{7 \pi}{6} = -\frac{\sqrt{3}}{2}
$$

So, the final answer is:
A. $$ \cos \frac{7 \pi}{6} = -\frac{\sqrt{3}}{2} $$
A. $$ \cos \frac{7 \pi}{6} = -\frac{\sqrt{3}}{2} $$

Upstudy AI Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Trigonometry Questions