Find the indicated function value. If it is undefined, say so. See Example 4. tan 450°

Verified step by step guidance

1. Recognize that the tangent function has a period of 180°, meaning \( \tan(\theta) = \tan(\theta + 180°) \). Use this property to simplify \( \tan 450° \) by subtracting multiples of 180° until the angle is within the standard range \( 0° \leq \theta < 360° \).
2. Calculate \( 450° - 360° = 90° \), so \( \tan 450° = \tan 90° \). This reduces the problem to finding \( \tan 90° \).
3. Recall the definition of tangent in terms of sine and cosine: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Substitute \( \theta = 90° \) to get \( \tan 90° = \frac{\sin 90°}{\cos 90°} \).
4. Evaluate \( \sin 90° \) and \( \cos 90° \). Since \( \sin 90° = 1 \) and \( \cos 90° = 0 \), the expression becomes \( \frac{1}{0} \), which is undefined because division by zero is not possible.
5. Conclude that \( \tan 450° \) is undefined because it simplifies to \( \tan 90° \), and tangent is undefined at 90° due to the cosine denominator being zero.

Key Concepts

Angle Measurement and Coterminal Angles

Angles greater than 360° can be simplified by subtracting 360° repeatedly to find a coterminal angle within the standard 0° to 360° range. This helps in evaluating trigonometric functions by reducing the angle to an equivalent, more manageable value.

Tangent Function Definition and Properties

The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. On the unit circle, tan(θ) = sin(θ)/cos(θ). Understanding where tangent is defined or undefined depends on the cosine value, as division by zero makes tangent undefined.

Evaluating Trigonometric Functions at Standard Angles

Certain angles like 0°, 30°, 45°, 60°, 90°, and their multiples have known sine and cosine values, which simplify tangent calculations. Recognizing these standard angles and their function values allows quick evaluation of trigonometric expressions.