Find each exact function value. See Example 3.
sin (-7π/ 6)

Verified step by step guidance

1. Recognize that the sine function is an odd function, which means \( \sin(-x) = -\sin(x) \). Therefore, \( \sin(-\frac{7\pi}{6}) = -\sin(\frac{7\pi}{6}) \).
2. Determine the reference angle for \( \frac{7\pi}{6} \). Since \( \frac{7\pi}{6} \) is in the third quadrant, the reference angle is \( \pi - \frac{7\pi}{6} = \frac{\pi}{6} \).
3. Recall that in the third quadrant, the sine function is negative. Therefore, \( \sin(\frac{7\pi}{6}) = -\sin(\frac{\pi}{6}) \).
4. Use the known value of \( \sin(\frac{\pi}{6}) \), which is \( \frac{1}{2} \).
5. Substitute back to find \( \sin(-\frac{7\pi}{6}) = -(-\frac{1}{2}) = \frac{1}{2} \).

Key Concepts

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is fundamental in trigonometry as it provides a geometric representation of the sine, cosine, and tangent functions. The angles measured in radians correspond to points on the circle, allowing for the determination of exact function values for various angles.

Reference Angles

A reference angle is the acute angle formed by the terminal side of a given angle and the x-axis. For angles in standard position, reference angles help simplify the calculation of trigonometric functions by relating them to angles within the first quadrant, where the values of sine and cosine are easier to determine.

Sine Function

The sine function, denoted as sin(θ), represents the ratio of the length of the opposite side to the hypotenuse in a right triangle. On the unit circle, it corresponds to the y-coordinate of a point at a given angle θ. Understanding the sine function is crucial for finding exact values, especially for angles that are not standard, such as negative angles or those greater than 2π.