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

Radians

Trigonometry

1. Measuring Angles

Radians

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Problem 1

Textbook Question

CONCEPT PREVIEW Find the exact length of each arc intercepted by the given central angle.

Verified step by step guidance

Step 1: Understand the problem. We need to find the length of an arc intercepted by a central angle. The formula to find the arc length (L) is L = r \theta, where r is the radius of the circle and \theta is the central angle in radians.

Step 2: Ensure the central angle is in radians. If the angle is given in degrees, convert it to radians using the conversion factor \frac{\pi}{180}.

Step 3: Identify the radius of the circle. This information should be provided in the problem or needs to be determined from the context.

Step 4: Substitute the radius and the central angle in radians into the arc length formula L = r \theta.

Step 5: Simplify the expression to find the exact length of the arc.

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Key Concepts

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

Central Angle

A central angle is an angle whose vertex is at the center of a circle and whose sides (rays) extend to the circumference. The measure of a central angle is directly related to the arc it intercepts, meaning that the angle helps determine the length of the arc. In a circle, the total measure of the angles is 360 degrees, and the central angle's proportion to this total can be used to find the length of the corresponding arc.

Arc Length Formula

The length of an arc can be calculated using the formula: Arc Length = (θ/360) × 2πr, where θ is the measure of the central angle in degrees and r is the radius of the circle. This formula derives from the relationship between the angle and the circumference of the circle, allowing for the determination of the arc's length based on the fraction of the circle that the angle represents.

Radians vs. Degrees

Angles can be measured in degrees or radians, with radians being the standard unit in many mathematical contexts. One complete revolution (360 degrees) is equivalent to 2π radians. When calculating arc lengths, it is essential to use the correct unit of measurement; if the angle is given in radians, the arc length formula simplifies to Arc Length = θ × r, making it crucial to convert between these units when necessary.