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In Exercises 49–54, find the measure of the side of the right triangle whose length is designated by a lowercase letter. Round answers to the nearest whole number.
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Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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TOC.tableOfContents
Capitolo 1, Problema 77
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: 4 inches Central Angle, θ: θ = 240°
Guida verificata passo dopo passo1
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.
Substitute the given values into the formula: \(r = 4\) inches and \(\theta = 240^\circ\), so the area becomes \(\frac{240}{360} \times \pi \times 4^2\).
Simplify the fraction \(\frac{240}{360}\) to its lowest terms to make calculations easier.
Calculate \$4^2$ to find the square of the radius.
Multiply the simplified fraction by \(\pi\) and the squared radius to express the area in terms of \(\pi\). Then, use the approximate value of \(\pi \approx 3.1416\) to round the area to two decimal places.
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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 using the formula (θ/360) × π × r², where θ is the central angle in degrees and r is the radius. This formula scales the full circle's area by the fraction of the circle represented by the angle.
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Percorso guidato
4:02
Calculating Area of SAS Triangles
Central Angle in Degrees
The central angle θ is the angle formed at the center of the circle by two radii. It determines the size of the sector and is measured in degrees. Understanding how to use this angle in formulas is essential for calculating sector properties like area and arc length.
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04:46Coterminal Angles
Rounding and Expressing Answers
After calculating the exact area in terms of π, it is often necessary to provide a numerical approximation. This involves substituting π ≈ 3.1416 and rounding the result to a specified number of decimal places, here two, to present a clear and practical answer.
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6:36Simplifying Trig Expressions
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