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 2

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 relationship between the central angle, radius, and arc length. The formula to find the arc length (L) is L = \theta \times r, where \theta is the central angle in radians and r is the radius of the circle.

Step 2: Ensure the central angle is in radians. If the angle is given in degrees, convert it to radians using the conversion factor: 1 degree = \frac{\pi}{180} radians.

Step 3: Identify the radius of the circle. This should be provided in the problem or assumed if not explicitly stated.

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

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 trigonometry, understanding central angles is crucial for calculating arc lengths and other properties of circles.

Arc Length Formula

The arc length of a circle can be calculated using the formula: Arc Length = r * θ, where 'r' is the radius of the circle and 'θ' is the central angle in radians. This formula shows the direct relationship between the radius, the angle, and the length of the arc. Converting degrees to radians may be necessary when using this formula, as it is commonly expressed in radians.

Radians and Degrees

Radians and degrees are two units for measuring angles. One complete revolution (360 degrees) is equivalent to 2π radians. Understanding how to convert between these two units is essential for solving problems involving angles in trigonometry, especially when applying formulas like the arc length formula, which typically requires angles to be in radians.