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

Evaluate Composite Trig Functions

Trigonometry

5. Inverse Trigonometric Functions and Basic Trigonometric Equations

Evaluate Composite Trig Functions

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1:14 minutes

Problem 39

Textbook Question

In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator.sin(sin⁻¹ 0.9)

Verified step by step guidance

Recognize that $ \sin^{-1} $ is the inverse sine function, also known as arcsine.

Understand that $ \sin(\sin^{-1}(x)) = x $ for $ x $ in the range $[-1, 1]$.

Since 0.9 is within the range $[-1, 1]$, the expression $ \sin(\sin^{-1}(0.9)) $ simplifies directly.

Apply the property of inverse functions: $ \sin(\sin^{-1}(0.9)) = 0.9 $.

Conclude that the exact value of the expression is simply 0.9.

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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. For example, sin⁻¹(0.9) gives the angle θ such that sin(θ) = 0.9. Understanding this concept is crucial for evaluating expressions involving inverse functions.

Sine Function

The sine function is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse. Knowing how to evaluate sin(θ) for specific angles or values is essential for solving trigonometric expressions.

Composition of Functions

Composition of functions involves applying one function to the result of another. In this case, evaluating sin(sin⁻¹(0.9)) means finding the sine of the angle whose sine is 0.9. This concept is important for simplifying expressions and understanding how functions interact.