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 5

Textbook Question

CONCEPT PREVIEW Find the measure of each central angle (in radians).

Verified step by step guidance

Step 1: Understand the concept of a central angle. A central angle is an angle whose vertex is at the center of a circle and whose sides are radii of the circle.

Step 2: Recall that the measure of a central angle in radians is the length of the arc divided by the radius of the circle. This is given by the formula: \( \theta = \frac{s}{r} \), where \( \theta \) is the central angle in radians, \( s \) is the arc length, and \( r \) is the radius.

Step 3: Identify the given values in the problem, such as the arc length \( s \) and the radius \( r \) of the circle.

Step 4: Substitute the given values into the formula \( \theta = \frac{s}{r} \) to find the measure of the central angle in radians.

Step 5: Simplify the expression to determine the measure of the central angle in radians.

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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 length it subtends on the circle. Understanding central angles is crucial for solving problems involving circular motion and geometry.

Radians

Radians are a unit of angular measure used in mathematics, particularly in trigonometry. One radian is defined as the angle formed when the arc length is equal to the radius of the circle. This unit is essential for converting between degrees and radians and is commonly used in calculus and physics for its natural relationship with circular motion.

Arc Length

Arc length is the distance along the curved line of a circle between two points. It can be calculated using the formula: arc length = radius × central angle (in radians). Understanding arc length is important for determining the relationship between angles and distances in circular geometry, which is often required in trigonometric problems.