Give all six trigonometric function values for each angle θ. Rationalize denominators when applicable. See Examples 5–7.
csc θ = ―3 , and cos θ > 0

Verified step by step guidance

1. Identify the given information: \( \csc \theta = -3 \) and \( \cos \theta > 0 \). Recall that \( \csc \theta = \frac{1}{\sin \theta} \), so use this to find \( \sin \theta \).
2. Calculate \( \sin \theta \) by taking the reciprocal of \( \csc \theta \): \( \sin \theta = \frac{1}{\csc \theta} = \frac{1}{-3} = -\frac{1}{3} \).
3. Determine the quadrant of \( \theta \) using the signs of \( \sin \theta \) and \( \cos \theta \). Since \( \sin \theta < 0 \) and \( \cos \theta > 0 \), \( \theta \) lies in the fourth quadrant.
4. Use the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to find \( \cos \theta \). Substitute \( \sin \theta = -\frac{1}{3} \) and solve for \( \cos \theta \), choosing the positive root because \( \cos \theta > 0 \).
5. Find the remaining trigonometric functions using the definitions: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), \( \cot \theta = \frac{1}{\tan \theta} \), \( \sec \theta = \frac{1}{\cos \theta} \), and rationalize denominators where necessary.

Key Concepts

Reciprocal Trigonometric Functions

The six trigonometric functions include sine, cosine, tangent, cosecant, secant, and cotangent. Cosecant (csc θ) is the reciprocal of sine (sin θ), so if csc θ = -3, then sin θ = -1/3. Understanding these reciprocal relationships is essential to find all function values.

Sign of Trigonometric Functions in Quadrants

The sign of trigonometric functions depends on the quadrant where the angle θ lies. Given csc θ = -3 (sin θ negative) and cos θ > 0 (cosine positive), θ must be in the fourth quadrant. Knowing quadrant signs helps determine the correct values and signs of all functions.

Pythagorean Identity and Rationalizing Denominators

The Pythagorean identity sin²θ + cos²θ = 1 allows calculation of unknown function values once one is known. After finding cos θ, tangent and other functions can be derived. Rationalizing denominators ensures answers are in simplified, standard form, improving clarity and precision.