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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2:05 minutes

Problem 72

Textbook Question

If θ is an acute angle and cos θ = 1/3, find csc (𝜋/2 - θ).

Verified step by step guidance

Recognize that \( \csc(\frac{\pi}{2} - \theta) \) is the cosecant of the complementary angle of \( \theta \).

Use the co-function identity: \( \csc(\frac{\pi}{2} - \theta) = \sec(\theta) \).

Recall that \( \sec(\theta) = \frac{1}{\cos(\theta)} \).

Substitute the given value of \( \cos(\theta) = \frac{1}{3} \) into the secant formula.

Calculate \( \sec(\theta) = \frac{1}{\frac{1}{3}} \) to find the value of \( \csc(\frac{\pi}{2} - \theta) \).

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Key Concepts

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

Trigonometric Functions

Trigonometric functions relate the angles of a triangle to the lengths of its sides. The primary functions include sine, cosine, and tangent, which are defined for acute angles in a right triangle. Understanding these functions is essential for solving problems involving angles and their relationships.

Cofunction Identity

Cofunction identities express the relationship between the sine and cosine of complementary angles. Specifically, csc(π/2 - θ) is equal to sin(θ). This identity is crucial for transforming expressions involving trigonometric functions and simplifying calculations.

Reciprocal Functions

Reciprocal functions in trigonometry include cosecant (csc), secant (sec), and cotangent (cot), which are the reciprocals of sine, cosine, and tangent, respectively. For example, csc(θ) = 1/sin(θ). Recognizing these relationships helps in finding values of trigonometric functions based on known values.