In Exercises 9–16, evaluate the trigonometric function at the quadrantal angle, or state that the expression is undefined. tan 𝜋

Key Concepts

Quadrantal Angles

Quadrantal angles are angles that are multiples of 90 degrees (or π/2 radians) and correspond to the axes in the Cartesian coordinate system. These angles include 0, π/2, π, 3π/2, and 2π. At these angles, the sine and cosine functions take on specific values, which are essential for evaluating trigonometric functions like tangent.

Tangent Function

The tangent function is defined as the ratio of the sine and cosine functions: tan(θ) = sin(θ) / cos(θ). For quadrantal angles, the values of sine and cosine can lead to specific outcomes, including undefined values when the cosine is zero. Understanding this ratio is crucial for evaluating the tangent at any angle, particularly at quadrantal angles.

Undefined Expressions in Trigonometry

In trigonometry, certain expressions can be undefined, particularly when they involve division by zero. For example, the tangent function is undefined at angles where the cosine is zero, such as π/2 and 3π/2. Recognizing when a trigonometric function is undefined is important for accurately interpreting and solving problems involving these functions.