Find each exact function value. See Example 2.
tan 3π/4

Verified step by step guidance

1. Recognize that \( \tan \theta \) is periodic with period \( \pi \), meaning \( \tan(\theta + \pi) = \tan \theta \).
2. Identify the reference angle for \( \frac{3\pi}{4} \). Since \( \frac{3\pi}{4} \) is in the second quadrant, the reference angle is \( \pi - \frac{3\pi}{4} = \frac{\pi}{4} \).
3. Recall that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). For \( \theta = \frac{\pi}{4} \), \( \tan \frac{\pi}{4} = 1 \).
4. Determine the sign of \( \tan \frac{3\pi}{4} \) based on the quadrant. In the second quadrant, tangent is negative.
5. Conclude that \( \tan \frac{3\pi}{4} = -\tan \frac{\pi}{4} = -1 \).

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, cosine, and tangent functions. Angles measured in radians correspond to points on the circle, allowing for the determination of exact function values for various angles.

Tangent Function

The tangent function, denoted as tan(θ), is defined as the ratio of the sine and cosine of an angle: tan(θ) = sin(θ) / cos(θ). It represents the slope of the line formed by the angle in the unit circle. Understanding how to calculate tangent values from the unit circle is essential for finding exact function values for specific angles.

Reference Angles

Reference angles are the acute angles formed by the terminal side of a given angle and the x-axis. They help simplify the calculation of trigonometric functions for angles greater than 90 degrees or less than 0 degrees. For example, the reference angle for 3π/4 is π/4, which allows us to find tan(3π/4) by using the known value of tan(π/4) and considering the sign based on the quadrant.