Ch. 3 - Radian Measure and The Unit Circle

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. 3 - Radian Measure and The Unit Circle

Problem 13

Chapter 4, Problem 13

Find the length to three significant digits of each arc intercepted by a central angle in a circle of radius r. See Example 1. r = 12.3 cm , θ = 2π/3 radians

Verified step by step guidance

Recall the formula for the length of an arc intercepted by a central angle \( \theta \) in a circle of radius \( r \): \[ \text{Arc length} = r \times \theta \]

Identify the given values from the problem: radius \( r = 12.3 \) cm and central angle \( \theta = \frac{2\pi}{3} \) radians.

Substitute the given values into the arc length formula: \[ \text{Arc length} = 12.3 \times \frac{2\pi}{3} \]

Multiply the radius by the fraction involving \( \pi \) to express the arc length in terms of \( \pi \): \[ \text{Arc length} = 12.3 \times \frac{2}{3} \times \pi \]

To find the numerical value, multiply \( 12.3 \times \frac{2}{3} \) first, then multiply the result by \( \pi \). Finally, round your answer to three significant digits.

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Key Concepts

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

Arc Length Formula

The arc length of a circle intercepted by a central angle is calculated using the formula s = rθ, where r is the radius and θ is the central angle in radians. This formula directly relates the angle measure to the length of the curved segment on the circle.

Radian Measure

Radians measure angles based on the radius of a circle, where one radian equals the angle subtended by an arc equal in length to the radius. Using radians simplifies arc length calculations since the formula s = rθ requires the angle in radians.

Significant Figures and Precision

When reporting numerical answers, significant figures indicate the precision of the measurement. Here, the arc length should be rounded to three significant digits, ensuring the result reflects the accuracy of the given data.

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Related Practice

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Use the formula ω = θ/t to find the value of the missing variable.

ω = 0.91 radian per min, t = 8.1 min

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Convert each degree measure to radians. Leave answers as multiples of π. See Examples 1(a) and 1(b). 30°

Textbook Question

Convert each degree measure to radians. Leave answers as multiples of π. See Examples 1(a) and 1(b). 90°

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Textbook Question

Convert each radian measure to degrees.

5π/4

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Textbook Question

Find the exact values of (a) sin s, (b) cos s, and (c) tan s for each real number s. See Example 1.

s = 2π

Textbook Question

Convert each degree measure to radians. Leave answers as multiples of π. See Examples 1(a) and 1(b). 60°

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