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 = 29.2 m, θ = 5π/6 radians
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 51
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
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 \(r = 30.0\) ft and central angle \(\theta = \frac{\pi}{2}\) radians.
Substitute the given values into the formula: \(\text{Area} = \frac{1}{2} \times (30.0)^{2} \times \frac{\pi}{2}\).
Simplify the expression step-by-step: first square the radius, then multiply by \(\frac{1}{2}\) and \(\frac{\pi}{2}\).
After simplifying, calculate the numerical value and round your answer to the nearest tenth to find 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 is the angle subtended by an arc equal in length to the radius. Using radians simplifies many trigonometric formulas, including the sector area formula, because it relates angle measure directly to arc length and area.
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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 understanding decimal places and applying standard rounding rules to present the answer clearly and accurately, especially in practical contexts like measurements.
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