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

3. Unit Circle

Trigonometric Functions on the Unit Circle

Trigonometry

3. Unit Circle

Trigonometric Functions on the Unit Circle

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2:14 minutes

Problem 25

Textbook Question

In Exercises 25–30, use an identity to find the value of each expression. Do not use a calculator.sin 37° csc 37°

Verified step by step guidance

Recognize that \( \csc \theta \) is the reciprocal of \( \sin \theta \), which means \( \csc \theta = \frac{1}{\sin \theta} \).

Substitute \( \csc 37^\circ \) with \( \frac{1}{\sin 37^\circ} \) in the expression \( \sin 37^\circ \cdot \csc 37^\circ \).

The expression becomes \( \sin 37^\circ \cdot \frac{1}{\sin 37^\circ} \).

Simplify the expression by canceling \( \sin 37^\circ \) in the numerator and the denominator.

Conclude that the simplified expression equals 1.

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Key Concepts

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

Sine Function

The sine function, denoted as sin(θ), is a fundamental trigonometric function that relates the angle θ of a right triangle to the ratio of the length of the opposite side to the hypotenuse. For example, sin(37°) represents this ratio for a triangle with a 37-degree angle.

Cosecant Function

The cosecant function, denoted as csc(θ), is the reciprocal of the sine function. It is defined as csc(θ) = 1/sin(θ). Therefore, csc(37°) is equal to the reciprocal of sin(37°), which means it represents the ratio of the hypotenuse to the opposite side in a right triangle.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. In this case, using the identity sin(θ) * csc(θ) = 1 can simplify the expression sin(37°) csc(37°) to 1, illustrating the relationship between sine and cosecant.