Trigonometry (11th Edition) Clone

Published by Pearson
ISBN 10: 978-0-13-421743-7
ISBN 13: 978-0-13421-743-7

Chapter 3 - Radian Measure and the Unit Circle - Section 3.2 Applications of Radian Measure - 3.2 Exercises - Page 113: 60


The central angle is $32.2^{\circ}$

Work Step by Step

Let $\theta$ be the angle in radians. Let $r$ be the radius. We can use the arc length $d$ to make an expression for the radius $r$: $d = \theta ~r$ $r = \frac{d}{\theta}$ Let $A$ be the area of the sector. Then the ratio of the angle $\theta$ to $2\pi$ is equal to the ratio of the sector area to the area of the whole circle. $\frac{\theta}{2\pi} = \frac{A}{\pi ~r^2}$ $\frac{\theta}{2\pi} = \frac{A}{\pi ~(\frac{d}{\theta})^2}$ $\frac{\theta}{2\pi} = \frac{A~\theta^2}{\pi ~d^2}$ $\theta = \frac{d^2}{2~A}$ $\theta = \frac{(6.0~cm)^2}{(2)(16~cm^2)}$ $\theta = 1.125~rad$ We can convert the angle $\theta$ to degrees: $\theta = (1.125~rad)(\frac{180^{\circ}}{2\pi~rad}) = 32.2^{\circ}$ The central angle is $32.2^{\circ}$
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