In Exercises 29–51, find the exact value of each expression. Do not use a calculator. tan[sin⁻¹ (− 1/2)]

Verified step by step guidance

1. Recognize that \( \sin^{-1}(-1/2) \) represents an angle \( \theta \) such that \( \sin(\theta) = -1/2 \).
2. Determine the range of \( \sin^{-1} \), which is \([-\pi/2, \pi/2]\), meaning \( \theta \) is in the fourth or first quadrant.
3. Since \( \sin(\theta) = -1/2 \) and \( \theta \) is in the fourth quadrant, identify \( \theta = -\pi/6 \) because \( \sin(-\pi/6) = -1/2 \).
4. Use the identity \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \) to find \( \tan(-\pi/6) \).
5. Recall that \( \cos(-\pi/6) = \sqrt{3}/2 \), so \( \tan(-\pi/6) = \frac{-1/2}{\sqrt{3}/2} = -\frac{1}{\sqrt{3}} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as sin⁻¹, are used to find angles when given a ratio. For example, sin⁻¹(-1/2) asks for the angle whose sine is -1/2. Understanding the range and output of these functions is crucial, as they can yield specific angles in different quadrants.

Trigonometric Ratios

Trigonometric ratios relate the angles of a triangle to the lengths of its sides. The tangent function, tan(θ), is defined as the ratio of the opposite side to the adjacent side in a right triangle. Knowing how to derive these ratios from the sine and cosine values is essential for solving the given expression.

Unit Circle

The unit circle is a fundamental concept in trigonometry that helps visualize the values of sine, cosine, and tangent for various angles. It provides a geometric interpretation of trigonometric functions, allowing for easy identification of angle values and their corresponding coordinates. This understanding is vital for evaluating expressions involving inverse trigonometric functions.