Textbook Question
In Exercises 5–7, convert each angle in radians to degrees. 5𝜋 3
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1. Measuring Angles
Radians
Problem 1
Textbook Question
CONCEPT PREVIEW Find the exact length of each arc intercepted by the given central angle.
Verified step by step guidance1
Identify the given central angle \( \theta \) and the radius \( r \) of the circle. The arc length depends on these two values.
Recall the formula for the length of an arc intercepted by a central angle:
\[ \text{Arc length} = r \times \theta \]
where \( \theta \) must be in radians.
If the central angle \( \theta \) is given in degrees, convert it to radians using the conversion:
\[ \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180} \]
Substitute the radius \( r \) and the central angle in radians \( \theta \) into the arc length formula to express the exact length of the arc.
Simplify the expression if possible, leaving the answer in terms of \( \pi \) for an exact value rather than a decimal approximation.
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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 intersect the circle, creating an arc. Understanding the measure of this angle is essential because it directly relates to the length of the intercepted arc.
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Arc Length Formula
The arc length of a circle segment is calculated using the formula: Arc Length = (θ/360) × 2πr, where θ is the central angle in degrees and r is the radius. This formula connects the angle measure to the portion of the circle's circumference.
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Radians and Degrees Conversion
Angles can be measured in degrees or radians, and converting between these units is often necessary. Since the arc length formula can use radians (Arc Length = rθ), knowing how to convert degrees to radians (radians = degrees × π/180) is crucial for accurate calculations.
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5:04Converting between Degrees & Radians
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