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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2:26 minutes

Problem 39

Textbook Question

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

Verified step by step guidance

Recognize that \( \sin^{-1} \left( \frac{\sqrt{2}}{2} \right) \) represents an angle \( \theta \) such that \( \sin \theta = \frac{\sqrt{2}}{2} \).

Recall that \( \sin \theta = \frac{\sqrt{2}}{2} \) corresponds to the angles \( \theta = \frac{\pi}{4} \) or \( \theta = \frac{3\pi}{4} \) in the unit circle.

Since \( \sin^{-1} \) (also known as arcsin) returns values in the range \(-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\), choose \( \theta = \frac{\pi}{4} \).

Now, find \( \cos(\theta) \) where \( \theta = \frac{\pi}{4} \).

Use the identity \( \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \) to determine the exact value.

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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, are used to find the angle whose sine is a given value. In this case, sin⁻¹(√2/2) yields an angle in the range of -π/2 to π/2, specifically π/4, since sin(π/4) = √2/2. Understanding how to interpret these functions is crucial for solving the problem.

Cosine Function

The cosine function relates the angle of a right triangle to the ratio of the length of the adjacent side to the hypotenuse. For any angle θ, cos(θ) can be found using the unit circle, where the x-coordinate of the point on the circle corresponding to θ gives the cosine value. This concept is essential for evaluating cos(sin⁻¹(√2/2)).

Pythagorean Identity

The Pythagorean identity states that for any angle θ, sin²(θ) + cos²(θ) = 1. This identity allows us to find the cosine of an angle if we know the sine. In this problem, once we determine sin(θ) = √2/2, we can use the identity to find cos(θ) by calculating cos(θ) = √(1 - sin²(θ)).