Evaluate each expression without using a calculator.
cos (tan⁻¹ (-2))

Verified step by step guidance

1. Recognize that \( \tan^{-1}(-2) \) represents an angle \( \theta \) such that \( \tan(\theta) = -2 \).
2. Visualize or draw a right triangle where the opposite side is -2 and the adjacent side is 1, since \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = -2 \).
3. Use the Pythagorean theorem to find the hypotenuse: \( \text{hypotenuse} = \sqrt{(-2)^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5} \).
4. Now, find \( \cos(\theta) \) using the definition of cosine: \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{1}{\sqrt{5}} \).
5. Rationalize the denominator if necessary: \( \cos(\theta) = \frac{1}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{5}}{5} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as tan⁻¹, are used to find angles when the value of a trigonometric function is known. For example, tan⁻¹(-2) gives the angle whose tangent is -2. Understanding how to interpret these functions is crucial for evaluating expressions involving them.

Right Triangle Relationships

Trigonometric functions are often defined in the context of right triangles. The tangent of an angle is the ratio of the opposite side to the adjacent side. When evaluating cos(tan⁻¹(-2)), it is helpful to visualize a right triangle where the tangent value corresponds to the given ratio, allowing for the calculation of the cosine.

Pythagorean Identity

The Pythagorean identity states that for any angle θ, sin²(θ) + cos²(θ) = 1. This identity is useful when calculating the cosine of an angle derived from an inverse function. By determining the sine and cosine values from the triangle formed by the tangent ratio, one can apply this identity to find the required cosine value.