Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 1305071751
ISBN 13: 978-1-30507-175-9

Chapter 5 - Section 5.1 - The Unit Circle - 5.1 Exercises - Page 408: 39

Answer

a. $\displaystyle \overline{t}=\frac{2\pi}{9}$ b. $\displaystyle \overline{t}=\frac{2\pi}{9}$ c. $\overline{t}\approx 0.14$ d. $\overline{t}\approx 1.28$

Work Step by Step

The reference number associated with the real number $t$ is the shortest distance along the unit circle between the terminal point determined by $t$ and the x-axis. For each t, find each terminal point on the unit circle (positive=counterclockwise) and associate it with the terminal point of some t between 0 and $ 2\pi$ If the terminal point "lands" in quadrants II, III or IV, choose the symmetric terminal number ($\pm\pi$) in quadrant I$:$ t in Q.II $\Rightarrow \overline{t}=\pi-t$ t in Q.III$\Rightarrow \overline{t}=t-\pi$ t in Q.IV$\Rightarrow \overline{t}=2\pi-t$ ------------------- a. The terminal point of $\displaystyle \frac{7\pi}{9}$ is in Q.II ($\pi= \displaystyle \frac{9\pi}{9}$), its reference number is $\pi- \displaystyle \frac{7\pi}{9}=\frac{2\pi}{9} $ b. The terminal point of $-\displaystyle \frac{7\pi}{9}$ is in Q.III (clockwise, $-\pi=- \displaystyle \frac{9\pi}{9}$), the same as the terminal point of $\displaystyle \frac{11\pi}{9}$ its reference number is $\displaystyle \frac{11\pi}{9}-\pi=\frac{2\pi}{9} $ c. The terminal point of $t=-3$ is in Q.III (clockwise, $-\pi\approx-3.14)$ is the same terminal point as $\approx\pi+0.14$ , its reference number is $\approx\pi+0.14-\pi=0.14$ d. The terminal point of 5 is in Q.III, (2$\pi\approx$6.28), its reference number is $2\pi-5\approx 1.28$
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