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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Problem 6.47

Textbook Question

Find the degree measure of θ if it exists. Do not use a calculator.
θ = sin⁻¹ 2

Verified step by step guidance

Understand that \( \sin^{-1} \) or \( \arcsin \) is the inverse sine function, which gives the angle whose sine is a given number.

Recognize that the range of the \( \sin^{-1} \) function is \([-\frac{\pi}{2}, \frac{\pi}{2}]\) or \([-90^\circ, 90^\circ]\).

Recall that the sine of an angle can only be between -1 and 1, inclusive.

Since 2 is outside the range of the sine function, there is no angle \( \theta \) such that \( \sin(\theta) = 2 \).

Conclude that \( \theta = \sin^{-1}(2) \) does not exist because 2 is not within the valid range for the sine function.

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Key Concepts

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

Inverse Sine Function

The inverse sine function, denoted as sin⁻¹ or arcsin, is used to find an angle whose sine value is a given number. The output of this function is restricted to the range of -90° to 90° (or -π/2 to π/2 radians), which means it can only return angles for sine values between -1 and 1. If the input exceeds this range, the function does not yield a valid angle.

Range of the Sine Function

The sine function outputs values between -1 and 1 for all real numbers. This means that for any angle θ, sin(θ) will always fall within this interval. Therefore, when evaluating sin⁻¹, the input must also lie within this range; otherwise, the function is undefined, indicating that no angle corresponds to the given sine value.

Understanding Undefined Values

In trigonometry, certain values can lead to undefined results. For instance, when attempting to find the angle θ such that sin(θ) = 2, this is impossible since 2 is outside the range of the sine function. Recognizing when a value is undefined is crucial for solving trigonometric equations and understanding the limitations of trigonometric functions.