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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3:08 minutes

Problem 63

Textbook Question

In Exercises 63–82, use a sketch to find the exact value of each expression.cos (sin⁻¹ 4/5)

Verified step by step guidance

Start by understanding that \( \sin^{-1} \left( \frac{4}{5} \right) \) represents an angle whose sine is \( \frac{4}{5} \).

Visualize a right triangle where the opposite side to the angle is 4 and the hypotenuse is 5.

Use the Pythagorean theorem to find the adjacent side: \( a^2 + 4^2 = 5^2 \).

Solve for the adjacent side \( a \) to find \( a = \sqrt{5^2 - 4^2} \).

Now, use the definition of cosine: \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \) to find \( \cos(\sin^{-1}(\frac{4}{5})) \).

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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. In this case, sin⁻¹(4/5) gives an angle θ such that sin(θ) = 4/5. Understanding how to interpret these functions is crucial for solving problems involving angles derived from trigonometric ratios.

Right Triangle Relationships

Trigonometric functions are often defined in the context of right triangles. For sin(θ) = opposite/hypotenuse, if sin(θ) = 4/5, we can visualize a right triangle where the opposite side is 4 and the hypotenuse is 5. This relationship allows us to find the adjacent side using the Pythagorean theorem, which is essential for calculating other trigonometric values.

Cosine Function

The cosine function relates the adjacent side of a right triangle to its hypotenuse. Once we determine the lengths of the sides of the triangle from the sine value, we can find cos(θ) using the formula cos(θ) = adjacent/hypotenuse. This step is necessary to evaluate the expression cos(sin⁻¹(4/5)) and find its exact value.