In Exercises 61–86, use reference angles to find the exact value of each expression. Do not use a calculator. tan(9𝜋/2)

Verified step by step guidance

1. Recognize that the angle given is \( \frac{9\pi}{2} \), which is greater than \( 2\pi \). Since trigonometric functions are periodic, find a coterminal angle by subtracting multiples of \( 2\pi \) until the angle lies within the interval \( [0, 2\pi) \). Use the formula: \( \theta_{coterminal} = \theta - 2\pi k \), where \( k \) is an integer.
2. Calculate the coterminal angle: \( \frac{9\pi}{2} - 2\pi \times 2 = \frac{9\pi}{2} - \frac{8\pi}{2} = \frac{\pi}{2} \). So, \( \frac{9\pi}{2} \) is coterminal with \( \frac{\pi}{2} \).
3. Identify the reference angle for \( \frac{\pi}{2} \). Since \( \frac{\pi}{2} \) lies on the positive y-axis, the reference angle is \( 0 \) because it is exactly on the axis.
4. Recall the value of \( \tan \) at \( \frac{\pi}{2} \). Since tangent is \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), and \( \cos \frac{\pi}{2} = 0 \), the tangent function is undefined at \( \frac{\pi}{2} \).
5. Conclude that \( \tan \frac{9\pi}{2} \) is undefined because it is coterminal with \( \tan \frac{\pi}{2} \), where tangent has a vertical asymptote.

Key Concepts

Reference Angles

A reference angle is the acute angle formed between the terminal side of an angle and the x-axis. It helps simplify trigonometric calculations by relating any angle to an angle between 0° and 90° (0 and π/2 radians). Using reference angles allows you to find exact trigonometric values without a calculator.

Angle Coterminality and Reduction

Angles that differ by full rotations (multiples of 2π radians) share the same terminal side and thus have the same trigonometric values. Reducing an angle like 9π/2 by subtracting multiples of 2π helps find a coterminal angle within one rotation, simplifying the evaluation of trigonometric functions.

Tangent Function Properties

The tangent function is periodic with period π, meaning tan(θ) = tan(θ + nπ) for any integer n. It is defined as the ratio of sine to cosine (tan θ = sin θ / cos θ) and can be positive or negative depending on the quadrant of the angle. Understanding its periodicity and sign is essential for finding exact values.