Question 5 If $\theta=\frac{-7 \pi}{6}$, then \[ \begin{array}{l} \sin (\theta)= \\ \cos (\theta)= \end{array} \] Give exact values. No decimals allowed! Example: Enter sqrt(2)/2 for $\frac{\sqrt{2}}{2}$. With functions like sqrt, be sure to use function notation (parentheses). sqrt(2)/2 will work, but sqrt $2 / 2$ will not. Calculator
Solution
Step 1 ：The question asks for the sine and cosine values of the angle \(\theta=-\frac{7 \pi}{6}\).
Step 2 ：The sine and cosine values of an angle can be found using the unit circle. The unit circle is a circle with a radius of 1 that is centered at the origin of the coordinate plane. The x-coordinate of a point on the unit circle represents the cosine of the angle formed by the positive x-axis and the line segment connecting the origin and the point. The y-coordinate represents the sine of the angle.
Step 3 ：The angle \(\theta=-\frac{7 \pi}{6}\) is equivalent to \(-\frac{7}{6} \times 180^\circ = -210^\circ\). This angle is in the third quadrant of the unit circle, where both sine and cosine are negative.
Step 4 ：The reference angle in the third quadrant is \(210^\circ - 180^\circ = 30^\circ\). The sine and cosine of \(30^\circ\) are \(\frac{1}{2}\) and \(\frac{\sqrt{3}}{2}\), respectively. Therefore, the sine and cosine of \(-210^\circ\) are \(-\frac{1}{2}\) and \(-\frac{\sqrt{3}}{2}\), respectively.
Step 5 ：Final Answer: The sine of \(\theta=-\frac{7 \pi}{6}\) is \(\boxed{\frac{1}{2}}\) and the cosine of \(\theta=-\frac{7 \pi}{6}\) is \(\boxed{-\frac{\sqrt{3}}{2}}\).
