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

5. Inverse Trigonometric Functions and Basic Trigonometric Equations

Inverse Sine, Cosine, & Tangent

Trigonometry

5. Inverse Trigonometric Functions and Basic Trigonometric Equations

Inverse Sine, Cosine, & Tangent

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1:58 minutes

Problem 51

Textbook Question

In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator.sin⁻¹ (sin π)

Verified step by step guidance

Recognize that \( \sin^{-1} \) is the inverse sine function, also known as arcsin, which returns the angle whose sine is the given value.

Understand that \( \sin(\pi) = 0 \) because the sine of \( \pi \) radians (or 180 degrees) is 0.

The expression \( \sin^{-1}(\sin(\pi)) \) simplifies to \( \sin^{-1}(0) \).

Recall that the range of \( \sin^{-1} \) is \([-\frac{\pi}{2}, \frac{\pi}{2}]\), meaning it returns angles within this interval.

Determine the angle within the range \([-\frac{\pi}{2}, \frac{\pi}{2}]\) whose sine is 0, which is 0.

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Key Concepts

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

Inverse Trigonometric Functions

Inverse trigonometric functions, such as sin⁻¹(x), are used to find the angle whose sine is x. These functions have specific ranges; for sin⁻¹(x), the output is restricted to the interval [-π/2, π/2]. Understanding this range is crucial for determining the correct angle when evaluating expressions involving inverse sine.

Unit Circle

The unit circle is a fundamental concept in trigonometry that defines the sine and cosine of angles based on their coordinates on a circle with a radius of one. The sine of an angle corresponds to the y-coordinate of the point on the unit circle. Knowing how to use the unit circle helps in visualizing and calculating the values of trigonometric functions for various angles.

Periodic Nature of Sine Function

The sine function is periodic with a period of 2π, meaning it repeats its values every 2π radians. This periodicity implies that sin(π) = 0, but when using the inverse sine function, we must consider the principal value. Thus, while sin(π) equals 0, sin⁻¹(0) yields a specific angle within the defined range of the inverse function.