In Exercises 55–58, use the given information to find the exact value of each of the following: α c. tan ------ 2 4 tan α = ------ , 180° < α < 270° 3

Key Concepts

Tangent Function

The tangent function, denoted as tan(θ), is a fundamental trigonometric function defined as the ratio of the opposite side to the adjacent side in a right triangle. It can also be expressed in terms of sine and cosine as tan(θ) = sin(θ)/cos(θ). Understanding the properties of the tangent function, including its periodicity and behavior in different quadrants, is essential for solving trigonometric equations.

Quadrants of the Unit Circle

The unit circle is divided into four quadrants, each corresponding to specific ranges of angle measures. In the third quadrant (180° < α < 270°), both sine and cosine values are negative, which affects the sign of the tangent function. Recognizing the quadrant in which an angle lies helps determine the sign of trigonometric values and is crucial for finding exact values.

Exact Values of Trigonometric Functions

Exact values of trigonometric functions can be derived from special angles (like 30°, 45°, and 60°) or by using known values and identities. For example, if tan(α) is given as a fraction, one can use the properties of triangles or the unit circle to find the exact value of tan(α/2) using the half-angle formula. Mastery of these techniques is vital for accurately solving trigonometric problems.