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

Verified step by step guidance

1. Recognize that the expression is \( \cos(\tan^{-1}(-2)) \). Here, \( \tan^{-1}(-2) \) represents an angle \( \theta \) whose tangent is \( -2 \). So, set \( \theta = \tan^{-1}(-2) \), which means \( \tan(\theta) = -2 \).
2. Visualize or draw a right triangle to represent the angle \( \theta \). Since \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = -2 \), assign the opposite side length as \( -2 \) and the adjacent side length as \( 1 \) (the negative sign indicates direction, but for length use positive values and consider the quadrant later).
3. Calculate the hypotenuse \( h \) of the triangle using the Pythagorean theorem: \( h = \sqrt{(\text{opposite})^2 + (\text{adjacent})^2} = \sqrt{(-2)^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5} \).
4. Recall that \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \). Using the triangle, this becomes \( \cos(\theta) = \frac{1}{\sqrt{5}} \).
5. Determine the sign of \( \cos(\theta) \) based on the quadrant of \( \theta \). Since \( \tan(\theta) = -2 \) is negative, \( \theta \) lies either in the second or fourth quadrant. Cosine is positive in the fourth quadrant and negative in the second. Because the tangent is negative and the angle is the inverse tangent of a negative number, \( \theta \) is in the fourth quadrant, so \( \cos(\theta) \) is positive. Therefore, \( \cos(\tan^{-1}(-2)) = \frac{1}{\sqrt{5}} \).

Key Concepts

Inverse Tangent Function (tan⁻¹ or arctan)

The inverse tangent function, tan⁻¹(x), returns the angle whose tangent is x. It maps a real number to an angle typically in the range (-π/2, π/2). Understanding this helps convert the given value into an angle for further trigonometric evaluation.

Right Triangle Interpretation of Trigonometric Ratios

Trigonometric functions can be interpreted using right triangles, where tangent is the ratio of the opposite side to the adjacent side. By representing tan⁻¹(-2) as an angle in a triangle, we can find the lengths of sides and use them to calculate cosine without a calculator.

Relationship Between Sine, Cosine, and Tangent

Cosine, sine, and tangent are related by the identity tan(θ) = sin(θ)/cos(θ). Knowing tan(θ), we can express cosine in terms of tangent using the Pythagorean identity cos(θ) = ±1/√(1 + tan²(θ)). This relationship allows evaluation of cosine from a given tangent value.