Ch. 1 - Angles and the Trigonometric Functions

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 1 - Angles and the Trigonometric Functions

Problem 77

Chapter 1, Problem 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°

Verified step by step guidance

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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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 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.

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.

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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