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

1. Measuring Angles

Angles in Standard Position

Trigonometry

1. Measuring Angles

Angles in Standard Position

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2:02 minutes

Problem 59

Textbook Question

In Exercises 59–62, use a calculator to find the value of the acute angle θ in radians, rounded to three decimal places.cos θ = 0.4112

Verified step by step guidance

Identify the trigonometric function involved: \( \cos \theta = 0.4112 \).

To find the angle \( \theta \), use the inverse cosine function: \( \theta = \cos^{-1}(0.4112) \).

Ensure your calculator is set to radians mode, as the problem requires the angle in radians.

Input the value 0.4112 into your calculator and apply the inverse cosine function to find \( \theta \).

Round the result to three decimal places to obtain the final value of \( \theta \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Cosine Function

The cosine function is a fundamental trigonometric function defined as the ratio of the adjacent side to the hypotenuse in a right triangle. It is denoted as cos(θ) and is crucial for determining the angle when the cosine value is known. In this context, cos(θ) = 0.4112 indicates the relationship between the angle θ and the lengths of the sides of a right triangle.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arccosine (cos⁻¹), are used to find angles when the value of a trigonometric function is known. For example, to find θ when cos(θ) = 0.4112, one would use θ = cos⁻¹(0.4112). This process is essential for solving for angles in trigonometric equations.

Radians

Radians are a unit of angular measure used in mathematics, particularly in trigonometry. One radian is the angle formed when the arc length is equal to the radius of the circle. In this problem, the angle θ needs to be expressed in radians, which is important for consistency in calculations and when using calculators that may default to radians.