Trigonometry (10th Edition)

Published by Pearson
ISBN 10: 0321671775
ISBN 13: 978-0-32167-177-6

Chapter 3 - Radian Measure and the Unit Circle - Section 3.1 Radian Measure - 3.1 Exercises - Page 98: 28

Answer

Let $t$ be the radian measure of an angle. The arc length is equal to $t~r$. When the radius is $r=1$, then the arc length is equal to $t$.

Work Step by Step

Let $t$ be the radian measure of an angle. One full rotation is $2\pi$ radians. The angle $t$ is a ratio of $\frac{t}{2\pi}$ of one full rotation. Let $r$ be the radius of the circle. One full rotation around the circle is the circumference of a circle which has an arc length of $2\pi~r$. When the ratio of one full rotation is $\frac{t}{2\pi}$, the arc length is $\frac{t}{2\pi}\times 2\pi~r$ which is equal to $t~r$. When the radius is $r=1$, then the arc length is equal to $t$.
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