Find each exact function value. See Example 3.
sin (-8π/ 3)

Verified step by step guidance

1. Recognize that the sine function is periodic with a period of \$2\pi\(. This means \)\sin(\theta) = \sin(\theta + 2\pi k)\( for any integer \)k$.
2. To find an equivalent angle for \(-\frac{8\pi}{3}\) within the standard range of \([0, 2\pi)\), add \$2\pi$ repeatedly until the angle is positive and within this range.
3. Calculate \(-\frac{8\pi}{3} + 2\pi = -\frac{8\pi}{3} + \frac{6\pi}{3} = -\frac{2\pi}{3}\). Since this is still negative, add \$2\pi$ again.
4. Add \$2\pi\( again: \)-\frac{2\pi}{3} + 2\pi = -\frac{2\pi}{3} + \frac{6\pi}{3} = \frac{4\pi}{3}\(. Now, this angle is within the range \)[0, 2\pi)$.
5. Use the unit circle to find \(\sin(\frac{4\pi}{3})\). The angle \(\frac{4\pi}{3}\) is in the third quadrant where sine is negative. The reference angle is \(\frac{\pi}{3}\), so \(\sin(\frac{4\pi}{3}) = -\sin(\frac{\pi}{3})\).

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 allows us to define the sine, cosine, and tangent functions for all angles. The coordinates of points on the unit circle correspond to the cosine and sine values of the angle formed with the positive x-axis.

Reference Angles

Reference angles are the acute angles formed by the terminal side of an angle and the x-axis. They help simplify the calculation of trigonometric functions for angles greater than 90 degrees or negative angles. For example, to find sin(-8π/3), we first convert it to a positive angle by adding 2π until it falls within the range of 0 to 2π, then find its reference angle.

Periodic Nature of Trigonometric Functions

Trigonometric functions are periodic, meaning they repeat their values in regular intervals. For sine and cosine, the period is 2π, while for tangent, it is π. This property allows us to find equivalent angles by adding or subtracting multiples of the period, which is essential when dealing with angles like -8π/3, as we can find a coterminal angle within the standard range.