Textbook Question
Convert each degree measure to radians. If applicable, round to the nearest thousandth. See Example 1(c).
42.5°
1
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Ch. 3 - Radian Measure and The Unit Circle
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 4, Problem 49
Find the area of a sector of a circle having radius r and central angle θ. Express answers to the nearest tenth. See Example 5. r = 29.2 m, θ = 5π/6 radians
Verified step by step guidance1
Recall the formula for the area of a sector of a circle: \(\text{Area} = \frac{1}{2} r^{2} \theta\), where \(r\) is the radius and \(\theta\) is the central angle in radians.
Identify the given values: radius \(r = 29.2\) meters and central angle \(\theta = \frac{5\pi}{6}\) radians.
Substitute the given values into the formula: \(\text{Area} = \frac{1}{2} \times (29.2)^{2} \times \frac{5\pi}{6}\).
Simplify the expression step-by-step: first calculate \(r^{2} = (29.2)^{2}\), then multiply by \(\frac{5\pi}{6}\), and finally multiply by \(\frac{1}{2}\).
After simplifying, round the final result to the nearest tenth to express the area of the sector.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Area of a Sector
The area of a sector of a circle is a portion of the circle's total area, determined by the central angle. It is calculated using the formula A = (1/2) * r² * θ, where r is the radius and θ is the central angle in radians. This formula directly relates the angle to the fraction of the circle's area.
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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 formulas in trigonometry and geometry, such as the sector area formula, which requires the angle to be in radians for direct application.
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Rounding and Approximation
After calculating the area, results often need to be rounded to a specified precision, such as the nearest tenth. This involves using decimal approximation of π and performing arithmetic carefully to ensure the final answer meets the required accuracy.
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Related Practice
Textbook Question
Convert each degree measure to radians. If applicable, round to the nearest thousandth. See Example 1(c).
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Textbook Question
Find a calculator approximation to four decimal places for each circular function value. cos (-0.2443)
Textbook Question
Convert each degree measure to radians. If applicable, round to the nearest thousandth. See Example 1(c).
174° 50'
5
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Textbook Question
Find the area of a sector of a circle having radius r and central angle θ. Express answers to the nearest tenth. See Example 5.
r = 30.0 ft, θ = π/2 radians
Textbook Question
Convert each degree measure to radians. If applicable, round to the nearest thousandth. See Example 1(c).
64.29°
1
views