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

Verified step by step guidance

1. Recall that the sine function is periodic with period \(2\pi\), so \(\sin(\theta) = \sin(\theta + 2k\pi)\) for any integer \(k\). This can help simplify the angle if needed.
2. Identify the reference angle for \(-\frac{7\pi}{6}\). Since the angle is negative, it means we rotate clockwise from the positive x-axis. To find a positive coterminal angle, add \(2\pi\) to \(-\frac{7\pi}{6}\): \(-\frac{7\pi}{6} + 2\pi = -\frac{7\pi}{6} + \frac{12\pi}{6} = \frac{5\pi}{6}\).
3. Recognize that \(\frac{5\pi}{6}\) is in the second quadrant, where sine values are positive. The reference angle for \(\frac{5\pi}{6}\) is \(\pi - \frac{5\pi}{6} = \frac{\pi}{6}\).
4. Use the known sine value for the reference angle \(\frac{\pi}{6}\), which is \(\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\).
5. Since \(\sin(\theta)\) is positive in the second quadrant, conclude that \(\sin\left(-\frac{7\pi}{6}\right) = \sin\left(\frac{5\pi}{6}\right) = \frac{1}{2}\).

Key Concepts

Unit Circle and Angle Measurement

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Angles are measured in radians, where 2π radians equal 360 degrees. Understanding how to locate angles like -7π/6 on the unit circle helps determine the corresponding sine value.

Reference Angles and Negative Angles

A reference angle is the acute angle formed between the terminal side of the given angle and the x-axis. Negative angles indicate clockwise rotation from the positive x-axis. Converting negative angles to positive coterminal angles simplifies finding exact trigonometric values.

Sine Function on the Unit Circle

The sine of an angle corresponds to the y-coordinate of the point where the terminal side of the angle intersects the unit circle. Knowing the sine values for common angles and their signs in different quadrants allows for exact evaluation of sine at angles like -7π/6.