Find each exact function value. ​See Example 3.​
cos 3π

Verified step by step guidance

1. Identify the angle: The given angle is \(3\pi\).
2. Convert the angle to degrees if necessary: \(3\pi\) radians is equivalent to \(540^\circ\).
3. Determine the reference angle: Since \(540^\circ\) is more than \(360^\circ\), subtract \(360^\circ\) to find the reference angle, which is \(180^\circ\).
4. Use the unit circle: The cosine of \(180^\circ\) is known from the unit circle.
5. Apply the cosine function: Since \(540^\circ\) is equivalent to \(180^\circ\) in terms of cosine, use the value of \(\cos(180^\circ)\).

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 and cosine functions. The coordinates of points on the unit circle correspond to the cosine and sine values of angles measured in radians, allowing for easy calculation of exact function values.

Cosine Function

The cosine function, denoted as cos(θ), represents the x-coordinate of a point on the unit circle corresponding to an angle θ. It is periodic with a period of 2π, meaning that cos(θ) = cos(θ + 2πn) for any integer n. Understanding the behavior of the cosine function is essential for finding exact values at specific angles, such as 3π.

Angle Measurement in Radians

In trigonometry, angles can be measured in degrees or radians, with radians being the standard unit in mathematical contexts. One complete revolution around the circle is 2π radians. When evaluating trigonometric functions, it is crucial to convert degrees to radians if necessary, as many trigonometric identities and values are derived based on radian measures.