Find each exact function value. See Example 2.
sin (4π/3)

Verified step by step guidance

1. Identify the reference angle for \(\frac{4\pi}{3}\). The reference angle is the angle in the first quadrant that corresponds to \(\frac{4\pi}{3}\). Since \(\frac{4\pi}{3}\) is in the third quadrant, subtract \(\pi\) to find the reference angle: \(\frac{4\pi}{3} - \pi = \frac{\pi}{3}\).
2. Determine the sine of the reference angle. The sine of \(\frac{\pi}{3}\) is known from the unit circle to be \(\frac{\sqrt{3}}{2}\).
3. Consider the sign of the sine function in the third quadrant. In the third quadrant, both sine and cosine are negative.
4. Apply the sign to the sine value of the reference angle. Since sine is negative in the third quadrant, \(\sin(\frac{4\pi}{3}) = -\sin(\frac{\pi}{3})\).
5. Combine the results to find the exact value: \(\sin(\frac{4\pi}{3}) = -\frac{\sqrt{3}}{2}\).

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 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

A reference angle is the acute angle formed by the terminal side of an angle and the x-axis. It is used to simplify the calculation of trigonometric functions for angles greater than 90 degrees or less than 0 degrees. For example, the reference angle for 4π/3 radians is π/3, which helps in determining the sine value.

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 like 4π/3, which lies in the third quadrant where sine values are negative.