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Expert-verified Found in: Page 479 ### Precalculus Mathematics for Calculus

Book edition 7th Edition
Author(s) James Stewart, Lothar Redlin, Saleem Watson
Pages 948 pages
ISBN 9781337067508 # Comparing a Triangle and a Sector of a Circle Two wood sticks and a metal rod, each of length 1, are connected to form a triangle with angle ${\mathbit{\theta }}_{\mathbf{1}}$ at the point P, as shown in the first figure below. The rod is then bent to form an arc of a circle with center P, resulting in a smaller angle ${\mathbit{\theta }}_{\mathbf{2}}$ at the point P, as shown in the second figure. Find ${\mathbit{\theta }}_{\mathbf{1}}$, ${\mathbit{\theta }}_{\mathbf{2}}$, and ${\mathbit{\theta }}_{\mathbf{1}}\mathbf{-}{\mathbit{\theta }}_{\mathbf{2}}$. The value for ${\theta }_{1}$ is 1.047 and ${\theta }_{2}$ is 1.

The value for ${\theta }_{1}-{\theta }_{2}$ is 0.47.

See the step by step solution

## Step 1. Given information.

Since, first figure is an equilateral triangle therefore each angle is ${60}^{\circ }$ i.e. $\frac{\pi }{3}$

Radius and sides is 1unit in case of both the figures.

## Step 2. Write the concept.

Equilateral triangle has angle equal to ${60}^{\circ }$

Therefore ${\theta }_{1}=\frac{\pi }{3}$

Central angle incase of a sector is given by:

$s=r{\theta }_{2}$

## Step 3. Determining the values.

${\theta }_{1}=\frac{\pi }{3}\phantom{\rule{0ex}{0ex}}{\theta }_{1}=1.047\phantom{\rule{0ex}{0ex}}{\theta }_{2}=\frac{s}{r}\phantom{\rule{0ex}{0ex}}{\theta }_{2}=\frac{1}{1}\phantom{\rule{0ex}{0ex}}{\theta }_{1}-{\theta }_{2}=1.047-1\phantom{\rule{0ex}{0ex}}=0.047$ ### Want to see more solutions like these? 