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 = 40.0 mi, θ = 135°
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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 57
Work each problem. See Example 5. Angle Measure Find the measure (in radians) of a central angle of a sector of area 16 in² a circle of radius 3.0 in.
Verified step by step guidance1
Recall the formula for the area of a sector of a circle: \(A = \frac{1}{2} r^{2} \theta\), where \(A\) is the area of the sector, \(r\) is the radius, and \(\theta\) is the central angle in radians.
Identify the given values: the area \(A = 16\) in² and the radius \(r = 3.0\) in.
Substitute the known values into the formula: \(16 = \frac{1}{2} \times (3.0)^{2} \times \theta\).
Simplify the expression on the right side: calculate \(\frac{1}{2} \times 9 = 4.5\), so the equation becomes \(16 = 4.5 \times \theta\).
Solve for \(\theta\) by dividing both sides of the equation by 4.5: \(\theta = \frac{16}{4.5}\).
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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 given by the formula A = (1/2) * r² * θ, where r is the radius and θ is the central angle in radians. This formula relates the sector's area directly to the angle, allowing calculation of one when the other is known.
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Radian Measure of Angles
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, especially those involving arc length and sector area.
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Solving for the Central Angle
To find the central angle θ when the sector area and radius are known, rearrange the sector area formula to θ = (2 * A) / r². This step involves algebraic manipulation and understanding the relationship between the variables.
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Related Practice
Textbook Question
Convert each radian measure to degrees. Write answers to the nearest minute. See Example 2(c).
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Textbook Question
Without using a calculator, decide whether each function value is positive or negative. (Hint: Consider the radian measures of the quadrantal angles, and remember that π ≈ 3.14.)
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Textbook Question
Without using a calculator, decide whether each function value is positive or negative. (Hint: Consider the radian measures of the quadrantal angles, and remember that π ≈ 3.14.)
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Textbook Question
Without using a calculator, decide whether each function value is positive or negative. (Hint: Consider the radian measures of the quadrantal angles, and remember that π ≈ 3.14.)
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Textbook Question
Convert each radian measure to degrees. Write answers to the nearest minute. See Example 2(c).
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