Work each problem.
Given tan x = -5⁄4, where π/2< x < π, use the trigonometric identities to find cot x, csc x and sec x.

Verified step by step guidance

1. Identify the quadrant where the angle \( x \) lies. Since \( \frac{\pi}{2} < x < \pi \), \( x \) is in the second quadrant, where sine is positive and cosine is negative.
2. Recall the given value: \( \tan x = -\frac{5}{4} \). Use the identity \( \tan x = \frac{\sin x}{\cos x} \) to express sine and cosine in terms of a right triangle with opposite side 5 and adjacent side -4 (negative because cosine is negative in the second quadrant).
3. Calculate the hypotenuse \( r \) using the Pythagorean theorem: \( r = \sqrt{(-4)^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41} \).
4. Find \( \sin x = \frac{\text{opposite}}{r} = \frac{5}{\sqrt{41}} \) and \( \cos x = \frac{\text{adjacent}}{r} = \frac{-4}{\sqrt{41}} \).
5. Use the definitions of the other trigonometric functions: \( \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x} \), \( \csc x = \frac{1}{\sin x} \), and \( \sec x = \frac{1}{\cos x} \). Substitute the values found for \( \sin x \) and \( \cos x \) to express \( \cot x \), \( \csc x \), and \( \sec x \).

Key Concepts

Trigonometric Ratios and Their Definitions

Trigonometric ratios relate the angles of a right triangle to the ratios of its sides. For any angle x, tan x is the ratio of the opposite side to the adjacent side. Cotangent (cot x) is the reciprocal of tangent, cosecant (csc x) is the reciprocal of sine, and secant (sec x) is the reciprocal of cosine.

Using the Pythagorean Identity

The Pythagorean identity states that sin²x + cos²x = 1 for any angle x. This identity helps find missing trigonometric values when one ratio is known. By expressing sine and cosine in terms of tangent, you can use this identity to calculate csc x and sec x.

Determining the Sign of Trigonometric Functions Based on Quadrants

The sign of trigonometric functions depends on the quadrant where the angle lies. Since π/2 < x < π (second quadrant), sine is positive, while cosine and tangent are negative. This information is crucial to assign correct signs to cot x, csc x, and sec x after calculation.