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 6.29

Textbook Question

Evaluate each expression without using a calculator.
sin (arccos (3/4))

Verified step by step guidance

Recognize that \( \arccos(3/4) \) represents an angle \( \theta \) such that \( \cos(\theta) = \frac{3}{4} \).

Visualize or draw a right triangle where the adjacent side to angle \( \theta \) is 3 and the hypotenuse is 4.

Use the Pythagorean theorem to find the opposite side: \( a^2 + 3^2 = 4^2 \), where \( a \) is the length of the opposite side.

Solve for \( a \) to find the length of the opposite side: \( a = \sqrt{4^2 - 3^2} \).

Now, use the definition of sine: \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{4} \).

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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 arccos, are used to find the angle whose cosine is a given value. For example, arccos(3/4) gives the angle θ such that cos(θ) = 3/4. Understanding how to interpret these functions is crucial for evaluating expressions involving them.

Pythagorean Identity

The Pythagorean identity states that for any angle θ, sin²(θ) + cos²(θ) = 1. This identity is essential when working with trigonometric functions, as it allows us to find the sine of an angle if we know its cosine. In this case, knowing cos(θ) = 3/4 helps us calculate sin(θ) using this identity.

Trigonometric Ratios

Trigonometric ratios relate the angles of a right triangle to the lengths of its sides. For a given angle θ, sin(θ) is defined as the ratio of the length of the opposite side to the hypotenuse. By applying these ratios, we can derive the sine value from the cosine value obtained from the inverse function.