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Expert-verified Found in: Page 408 ### Precalculus Mathematics for Calculus

Book edition 7th Edition
Author(s) James Stewart, Lothar Redlin, Saleem Watson
Pages 948 pages
ISBN 9781337067508 # Find (a) the reference number for each value of $t$ and (b) the terminal point determined by $t$. $t=-\frac{11\pi }{3}$

a. The reference number $\overline{t}=\frac{\pi }{3}$.

b. The terminal point for $t=-\frac{11\pi }{3}$ is $P\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)$.

See the step by step solution

## Part a. Step 1. State the definition of the reference number.

The reference number $\overline{t}$ associated with $t$, a real number is the shortest distance along the unit circle between the terminal point determined by $t$ and the $x$-axis.

## Part a. Step 2. Determine the quadrant where the terminal point lies.

If $0, then $t$ lies in quadrant I and the reference number $\overline{t}=t$.

If $\frac{\pi }{2}, then $t$ lies in quadrant II and the reference number $\overline{t}=\pi -t$.

If $\pi , then $t$ lies in quadrant III and the reference number $\overline{t}=t-\pi$.

If $\frac{3\pi }{2}, then $t$ lies in quadrant IV and the reference number $\overline{t}=2\pi -t$.

Here, $t=-\frac{11\pi }{3}$, which lies in quadrant I.

## Part a. Step 3. Find the reference number.

Since $t$ lies in quadrant I, therefore, the reference number will be $\overline{t}=4\pi +t$. Here substitute $-\frac{11\pi }{3}$ for $t$ and simplify for $\overline{t}$.

$\overline{t}=4\pi +t\phantom{\rule{0ex}{0ex}}=4\pi +\left(-\frac{11\pi }{3}\right)\phantom{\rule{0ex}{0ex}}=\frac{\pi }{3}$

Therefore, the reference number $\overline{t}=\frac{\pi }{3}$.

## Part b. Step 1. State the definition of the terminal point on the unit circle.

Starting at the point $\left(1,0\right)$, if $t\ge 0$, then $t$ is the distance along the unit circle in clockwise direction and if $t<0$, then $\left|t\right|$ is the distance along the unit circle in clockwise direction. Here, $t$ is a real number and this distance generates a point $P\left(x,y\right)$ on the unit circle. The point $P\left(x,y\right)$ obtained in this way is called the terminal point determined by the real number $t$.

## Part b. Step 2. State the terminal points for some special values of t for reference.

The table below gives the terminal points for some special values of $t$.

 $t$ Terminal point determined by $t$ 0 $\left(1,0\right)$ $\frac{\pi }{6}$ $\left(\frac{\sqrt{3}}{2},\frac{1}{2}\right)$ $\frac{\pi }{4}$ $\left(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)$ $\frac{\pi }{3}$ $\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)$ $\frac{\pi }{2}$ $\left(0,1\right)$

## Part b. Step 3. Simplify and state the conclusion.

Let $P\left(x,y\right)$ be the terminal point for $t=-\frac{11\pi }{3}$.

Since the reference number for $t=-\frac{11\pi }{3}$ is $\overline{t}=\frac{\pi }{3}$ and the corresponding terminal point for $\overline{t}=\frac{\pi }{3}$ is $\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)$ in the first quadrant, note that $t=-\frac{11\pi }{3}$ lies in the quadrant I, therefore, its terminal point will have positive $x$-coordinate and positive $y$-coordinate. Therefore, the terminal point for $t=-\frac{11\pi }{3}$ is $P\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)$. ### Want to see more solutions like these? 