Find each exact function value. See Example 2. sin 7π/6

Verified step by step guidance

1. Identify the angle given: \(\frac{7\pi}{6}\). This angle is in radians and is greater than \(\pi\), so it lies in the third quadrant of the unit circle.
2. Recall that the sine function is negative in the third quadrant because sine corresponds to the y-coordinate on the unit circle, which is negative there.
3. Find the reference angle for \(\frac{7\pi}{6}\). The reference angle is the acute angle formed with the x-axis, calculated as \(\frac{7\pi}{6} - \pi = \frac{7\pi}{6} - \frac{6\pi}{6} = \frac{\pi}{6}\).
4. Use the known sine value of the reference angle \(\frac{\pi}{6}\), which is \(\sin \frac{\pi}{6} = \frac{1}{2}\).
5. Apply the sign based on the quadrant: since \(\frac{7\pi}{6}\) is in the third quadrant where sine is negative, \(\sin \frac{7\pi}{6} = -\sin \frac{\pi}{6} = -\frac{1}{2}\).

Key Concepts

Unit Circle and Radian Measure

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Angles measured in radians correspond to points on this circle, where the angle's measure represents the length of the arc subtended. Understanding the position of 7π/6 radians on the unit circle helps determine the sine value.

Sine Function on the Unit Circle

The sine of an angle in the unit circle is the y-coordinate of the corresponding point on the circle. For angles beyond π (180°), sine values can be negative, reflecting the point's position below the x-axis. Knowing this helps find the exact sine value for 7π/6.

Reference Angles

A reference angle is the acute angle formed between the terminal side of the given angle and the x-axis. It simplifies finding trigonometric values by relating them to known angles in the first quadrant. For 7π/6, the reference angle is π/6, which aids in determining the exact sine value.