In Exercises 29–51, find the exact value of each expression. Do not use a calculator. tan [cos⁻¹ (− 4/5)]

Verified step by step guidance

1. Recognize that \( \cos^{-1}(-\frac{4}{5}) \) represents an angle \( \theta \) such that \( \cos(\theta) = -\frac{4}{5} \).
2. Since \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \), assign the adjacent side as \(-4\) and the hypotenuse as \(5\).
3. Use the Pythagorean theorem to find the opposite side: \( \text{opposite} = \sqrt{5^2 - (-4)^2} = \sqrt{25 - 16} = \sqrt{9} = 3 \).
4. Determine the sign of the opposite side. Since \( \cos(\theta) \) is negative, \( \theta \) is in the second quadrant where \( \sin(\theta) \) is positive, so the opposite side is \(3\).
5. Calculate \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{3}{-4} = -\frac{3}{4} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as cos⁻¹, are used to find the angle whose cosine is a given value. In this case, cos⁻¹(−4/5) gives an angle θ such that cos(θ) = −4/5. Understanding how to interpret these functions is crucial for solving problems involving angles derived from trigonometric ratios.

Tangent Function

The tangent function, defined as the ratio of the opposite side to the adjacent side in a right triangle, is essential for evaluating expressions involving angles. For any angle θ, tan(θ) = sin(θ)/cos(θ). In this problem, once the angle from the inverse cosine is determined, the tangent can be calculated using the sine and cosine values associated with that angle.

Pythagorean Identity

The Pythagorean identity states that for any angle θ, sin²(θ) + cos²(θ) = 1. This identity is useful for finding the sine value when the cosine value is known. In this case, knowing cos(θ) = −4/5 allows us to find sin(θ) using the identity, which is necessary to compute tan(θ) accurately.