Textbook Question
In Exercises 59–62, use a calculator to find the value of the acute angle θ in radians, rounded to three decimal places. cos θ = 0.4112
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Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
Chapter 1, Problem 1.1.75
In Exercises 75–78, find the area of the sector of a circle of radius r formed by a central angle θ. Express area in terms of π. Then round your answer to two decimal places. Radius, r: 10 meters Central Angle, θ: θ = 18°
Verified step by step guidance1
Recall the formula for the area of a sector of a circle: \(\text{Area} = \frac{\theta}{360^\circ} \times \pi r^2\), where \(\theta\) is the central angle in degrees and \(r\) is the radius of the circle.
Substitute the given values into the formula: \(r = 10\) meters and \(\theta = 18^\circ\), so the area becomes \(\frac{18}{360} \times \pi \times 10^2\).
Simplify the fraction \(\frac{18}{360}\) to its lowest terms to make calculations easier.
Calculate the expression \(\pi \times 10^2\) which represents the area of the full circle, then multiply by the simplified fraction to find the sector area in terms of \(\pi\).
Finally, use the approximate value of \(\pi \approx 3.1416\) to compute the numerical value of the sector area and round your answer to two decimal places.
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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 the portion of the circle's area enclosed by two radii and the arc between them. It is calculated as (θ/360) × π × r² when θ is in degrees, where r is the radius and θ is the central angle.
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Central Angle in Degrees
The central angle θ is the angle formed at the center of the circle by two radii. When given in degrees, it must be used directly in the sector area formula as a fraction of 360°, representing the full circle.
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Rounding Numerical Results
After calculating the exact area in terms of π, numerical approximation involves substituting π ≈ 3.1416 and rounding the final answer to the specified decimal places, here two decimals, to provide a practical and understandable result.
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Related Practice
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Textbook Question
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