Find the exact value of each expression.
tan 285°

Verified step by step guidance

1. First, recognize that \( 285^\circ \) can be expressed as \( 360^\circ - 75^\circ \). This means we can use the identity for tangent of a difference: \( \tan(360^\circ - \theta) = -\tan(\theta) \).
2. Apply the identity: \( \tan(285^\circ) = \tan(360^\circ - 75^\circ) = -\tan(75^\circ) \).
3. Next, express \( 75^\circ \) as a sum of angles whose tangent values are known: \( 75^\circ = 45^\circ + 30^\circ \).
4. Use the tangent sum identity: \( \tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} \), where \( \alpha = 45^\circ \) and \( \beta = 30^\circ \).
5. Substitute the known values: \( \tan(45^\circ) = 1 \) and \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), then calculate \( \tan(75^\circ) \) using the formula, and finally find \( \tan(285^\circ) = -\tan(75^\circ) \).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry that defines the relationship between angles and the coordinates of points on a circle with a radius of one. Each angle corresponds to a point on the circle, where the x-coordinate represents the cosine of the angle and the y-coordinate represents the sine. Understanding the unit circle is essential for evaluating trigonometric functions at various angles, including those greater than 360°.

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 in different quadrants. For example, to find tan 285°, we can determine its reference angle, which is 360° - 285° = 75°, and then use the properties of tangent in the fourth quadrant to find the exact value.

Tangent Function

The tangent function is defined as the ratio of the sine to the cosine of an angle, or tan(θ) = sin(θ) / cos(θ). It is periodic with a period of 180°, meaning that tan(θ) = tan(θ + 180°). Understanding how to compute the tangent of an angle using the unit circle and reference angles is crucial for finding exact values, especially for angles like 285° that are not commonly found in basic trigonometric tables.