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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2:45 minutes

Problem 32

Textbook Question

In Exercises 29–51, find the exact value of each expression. Do not use a calculator._sin⁻¹ (− √3/2)

Verified step by step guidance

Recognize that \( \sin^{-1}(x) \) represents the angle whose sine is \( x \).

Identify that \( -\frac{\sqrt{3}}{2} \) is a known sine value corresponding to specific angles.

Recall that the sine function is negative in the third and fourth quadrants.

Determine the reference angle for \( \sin^{-1}(\frac{\sqrt{3}}{2}) \), which is \( \frac{\pi}{3} \) or 60 degrees.

Find the angle in the fourth quadrant where sine is negative, which is \( -\frac{\pi}{3} \) or \( -60 \) degrees.

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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 (also known as arcsin), are used to find the angle whose sine is a given value. For example, sin⁻¹(x) returns the angle θ such that sin(θ) = x. The range of arcsin is restricted to [-π/2, π/2] to ensure it is a function, meaning it can only return one value for each input.

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is fundamental in trigonometry as it provides a geometric interpretation of the sine, cosine, and tangent functions. The coordinates of points on the unit circle correspond to the cosine and sine values of angles, allowing for easy identification of exact values for trigonometric functions.

Reference Angles

Reference angles are the acute angles formed by the terminal side of an angle and the x-axis. They are crucial for determining the sine, cosine, and tangent values of angles in different quadrants. For negative sine values, such as sin(θ) = -√3/2, the reference angle can help identify the corresponding angle in the fourth quadrant, where sine is negative.