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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2:10 minutes

Problem 1

Textbook Question

In Exercises 1–26, find the exact value of each expression.sin⁻¹ 1/2

Verified step by step guidance

Recognize that $ \sin^{-1} $ is the inverse sine function, also known as arcsin, which gives the angle whose sine is a given number.

Identify that you need to find an angle $ \theta $ such that $ \sin(\theta) = \frac{1}{2} $.

Recall the unit circle and the common angles where the sine value is $ \frac{1}{2} $.

Note that $ \sin(\theta) = \frac{1}{2} $ at $ \theta = \frac{\pi}{6} $ and $ \theta = \frac{5\pi}{6} $ in the interval $[0, \pi]$.

Since $ \sin^{-1} $ returns the principal value, which is in the range $[-\frac{\pi}{2}, \frac{\pi}{2}]$, the exact value is $ \frac{\pi}{6} $.

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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 (arcsin), are used to find the angle whose sine is a given value. For example, sin⁻¹(1/2) asks for the angle θ such that sin(θ) = 1/2. The range of the arcsin function is limited to [-π/2, π/2], ensuring that each input corresponds to a unique output.

Unit Circle

The unit circle is a fundamental concept in trigonometry, representing all possible angles and their corresponding sine and cosine values. It is a circle with a radius of one centered at the origin of a coordinate plane. Understanding the unit circle helps in visualizing the values of sine and cosine for various angles, including those that yield sin(θ) = 1/2.

Special Angles

Special angles are commonly used angles in trigonometry, such as 0°, 30°, 45°, 60°, and 90°, which have known sine, cosine, and tangent values. For instance, sin(30°) = 1/2, which directly relates to the question. Recognizing these angles allows for quick identification of exact values in trigonometric expressions.