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

Verified step by step guidance

1. Recognize that the angle given is a quadrantal angle, specifically \(\pi\) radians, which corresponds to 180 degrees on the unit circle.
2. Recall that the tangent function is defined as the ratio of sine to cosine: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).
3. Evaluate \(\sin \pi\) and \(\cos \pi\) using the unit circle values: \(\sin \pi = 0\) and \(\cos \pi = -1\).
4. Substitute these values into the tangent formula: \(\tan \pi = \frac{0}{-1}\).
5. Simplify the fraction to find the value of \(\tan \pi\), noting that division by a nonzero number is defined.

Key Concepts

Quadrantal Angles

Quadrantal angles are angles that lie on the x- or y-axis in the coordinate plane, typically multiples of 90° or π/2 radians. These angles include 0, π/2, π, 3π/2, and 2π, where trigonometric functions often take special values or become undefined.

Tangent Function at Quadrantal Angles

The tangent function is defined as the ratio of sine to cosine (tan θ = sin θ / cos θ). At quadrantal angles, since cosine or sine can be zero, the tangent may be zero, a finite number, or undefined if division by zero occurs.

Evaluating Trigonometric Functions Using the Unit Circle

The unit circle provides coordinates (cos θ, sin θ) for any angle θ. Evaluating trigonometric functions at quadrantal angles involves identifying these coordinates and applying definitions, which helps determine exact values or identify undefined expressions.