Table of contents

0. Review of College Algebra4h 43m
1. Measuring Angles40m
2. Trigonometric Functions on Right Triangles2h 5m
3. Unit Circle1h 19m
4. Graphing Trigonometric Functions1h 19m
5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
6. Trigonometric Identities and More Equations2h 34m
7. Non-Right Triangles1h 38m
8. Vectors2h 25m
9. Polar Equations2h 5m
10. Parametric Equations1h 6m
11. Graphing Complex Numbers1h 7m

2. Trigonometric Functions on Right Triangles

Trigonometric Functions on Right Triangles

Trigonometry

2. Trigonometric Functions on Right Triangles

Trigonometric Functions on Right Triangles

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2:46 minutes

Problem 71

Textbook Question

Use identities to solve each of the following. Rationalize denominators when applicable. See Examples 5–7. Find cot θ , given that csc θ = ―1.45 and θ is in quadrant III.

Verified step by step guidance

Recall that \( \csc \theta = \frac{1}{\sin \theta} \). Given \( \csc \theta = -1.45 \), solve for \( \sin \theta \) by taking the reciprocal: \( \sin \theta = \frac{1}{-1.45} \).

Since \( \theta \) is in quadrant III, both sine and cosine are negative. Use the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to find \( \cos \theta \). Substitute \( \sin \theta \) into the identity and solve for \( \cos \theta \).

Calculate \( \cos \theta \) using the identity: \( \cos \theta = -\sqrt{1 - \sin^2 \theta} \). Remember to take the negative square root because \( \cos \theta \) is negative in quadrant III.

Recall that \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). Substitute the values of \( \cos \theta \) and \( \sin \theta \) into this formula.

Rationalize the denominator if necessary by multiplying the numerator and the denominator by the conjugate of the denominator.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Cosecant and Cotangent Functions

Cosecant (csc) is the reciprocal of sine, defined as csc θ = 1/sin θ. Cotangent (cot) is the reciprocal of tangent, defined as cot θ = cos θ/sin θ. Understanding these relationships is crucial for solving trigonometric problems, especially when given one function and needing to find another.

Quadrants and Sign of Trigonometric Functions

The unit circle is divided into four quadrants, each affecting the signs of trigonometric functions. In quadrant III, both sine and cosine are negative, which means cosecant and cotangent will also be negative. Recognizing the quadrant helps determine the signs of the trigonometric values involved in the problem.

Rationalizing Denominators

Rationalizing the denominator involves eliminating any radical expressions from the denominator of a fraction. This is often done by multiplying the numerator and denominator by a suitable expression. In trigonometry, this technique can simplify expressions and make calculations clearer, especially when dealing with trigonometric identities.