4.4 Chapter 4 Summary and Review

The London Eye, the world's largest Ferris wheel, completes one revolution every 30 minutes. By how many degrees will it rotate in 1 minute?

The London Eye in Problem 1 has 32 cabins evenly spaced along the wheel. If the cabins are numbered consecutively from 1 to 32, what is the angular separation between cabins number 1 and number 15?

1. ${510}^{\circ }$
2. ${600}^{\circ }$
3. $-{200}^{\circ }$
4. $-{700}^{\circ }$

1. $-{380}^{\circ }$
2. $-{423}^{\circ }$
3. ${187}^{\circ }$
4. ${1000}^{\circ }$

1. $-{300}^{\circ }$
2. $-{25}^{\circ }$
3. ${100}^{\circ }$
4. ${250}^{\circ }$

1. ${430}^{\circ }$
2. $-{590}^{\circ }$
3. $-{95}^{\circ }$
4. ${1050}^{\circ }$

Let $\tilde{\theta }=f(\theta )$ be the function that gives the reference angle of $\theta$. For example, $f({110}^{\circ })={70}^{\circ }$ because the reference angle for ${110}^{\circ }$ is ${70}^{\circ }$.

1. Fill in the table of values.

   $\theta$	$0^{\circ }$	${30}^{\circ }$	${60}^{\circ }$	${90}^{\circ }$	${120}^{\circ }$	${150}^{\circ }$	${180}^{\circ }$	${210}^{\circ }$	${240}^{\circ }$	${270}^{\circ }$	${300}^{\circ }$	${330}^{\circ }$	${360}^{\circ }$
   $f(\theta )$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$

2. Choose appropriate scales for the axes and graph the function for $-{360}^{\circ }\le \theta \le {360}^{\circ }$.

   

Let $\tilde{\theta }=f(\theta )$ be the function that gives the reference angle of $\theta$. (See Problem 7.) Is $f$ a periodic function? If so, give its period, midline, and amplitude. If not, explain why not.

$$\sin \theta ={\displaystyle \frac{-1}{2}}$$

$$\cos \theta ={\displaystyle \frac{-1}{\sqrt{2}}}$$

$$2\cos \theta +1=0$$

$$5\sin \theta +5=0$$

$$\tan \theta -1=0$$

$$\sqrt{3}+3\tan \theta =0$$

$$\cos \theta =\cos (-{23}^{\circ })$$

$$\sin \theta =\sin ({370}^{\circ })$$

$$\tan \theta =\tan {432}^{\circ }$$

$$\tan \theta =\tan (-6^{\circ })$$

$$\sin \theta +\sin {83}^{\circ }=0$$

$$\cos \theta +\cos {429}^{\circ }=0$$

$$3\sin \theta +2=0$$

$$5\cos \theta +4=0$$

$${\displaystyle \frac{2}{3}}\tan \theta +1=0$$

$$-4\tan \theta +12=0$$

$$4=8\cos \theta +9$$

$$-3=6\sin \theta -5$$

$$\theta ={193}^{\circ },r=10$$

$$\theta =-{12}^{\circ },r=20$$

$$\theta ={92}^{\circ },r=8$$

$$\theta ={403}^{\circ },r=6$$

$$\theta =-{341}^{\circ },r=3$$

$$\theta =-{107}^{\circ },r=20$$

If you follow a bearing of ${190}^{\circ }$ for 10 miles, how far south and how far west are you from your starting point? Round to two decimal places.

A child releases a balloon, which then follows a bearing of ${84}^{\circ }$ for 150 meters. How far east and how far north is the balloon from where it was released? Round to the nearest meter.

Amplitude 7, midline $y=4$, period 2, $y$-intercept 4

Amplitude 100, midline $y=50$, period 12, $y$-intercept 100

Maximum point at $({90}^{\circ },24)$ and minimum point at $({270}^{\circ },10)$

Horizontal intercept at ${180}^{\circ }$, maximum points at $(0^{\circ },5)$ and $({360}^{\circ },5)$

$2f(\theta )g(\theta )$ for $\theta ={30}^{\circ }$

$f\left({\displaystyle \frac{\theta }{2}}\right)$ for $\theta ={120}^{\circ }$

$g(4\theta )-f(2\theta )$ for $\theta ={15}^{\circ }$

${\displaystyle \frac{1-g(2\theta )}{2}}$ for $\theta ={45}^{\circ }$

Every 24 hours Delbert takes 50 mg of a therapeutic drug. The level of that drug in Delbert's bloodstream immediately jumps to its peak level of 60 mg, but the level diminishes to its lowest level of 10 mg just before the next dose.

1. Sketch a graph of $d(t)$, the amount of the drug in Delbert's bloodstream after one of his doses, Show at least three doses.
2. What is the period of $d(t)$?

A water fountain has water trickling into a container, but once the container is full, it tilts and the water pours quickly out. Then the container tilts back and starts to fill again. The container is filled 5 times every minute.

1. Sketch a graph of $h(t)$, the height of water in the container at time Show at least two complete fill-and-empty cycles.
2. What is the period of $h(t)$?

Henry is watching Billie ride a carousel. He stands 2 meters from the carousel, which has a diameter of 10 meters. He notices that she passes by him three times each minute.

1. Sketch a graph of $f(t)$, the distance between Henry and Billie at time $t$. Show at least two complete circuits.
2. What is the period of $f(t)$?

An ant walks at constant unit speed (1 unit of distance per second) along the triangle with vertices at $(0,0),(1,1),$ and $(0,1)$.

1. How far does the ant need to walk to get from $(0,0)$ to $(1,1)$? From $(1,1)$ to $(0,1)$? From $(0,1)$ to $(0,0)$?
2. Sketch a graph of $g(t)$, the ant's $y$-coordinate at time $t$. Show at least two circuits around the triangle.
3. What is the period of $g(t)$?

$$y=4+2\cos \theta$$

$$y=-1+3\cos \theta$$

$$y=1.5+3.5\sin 2\theta$$

$$y=1.6+1.4\sin (0.5\theta )$$

$$y={\displaystyle \frac{\sqrt{3}}{3}}x+1$$

$$y=-x-11$$

$$y=100-28x$$

$$y=-3.7+1.4x$$

$(0,2)$, $\alpha ={45}^{\circ }$

$(4,0)$, $\alpha ={135}^{\circ }$

$(3,-4)$, $\alpha ={120}^{\circ }$

$(-7,2)$, $\alpha ={60}^{\circ }$

Sketch the graphs of $y=\tan \theta$ and $y=\cos \theta$ on the same grid for $-{180}^{\circ }<\theta <{180}^{\circ }$. How are the $\theta$-intercepts of the graph of $y=\cos \theta$ related to the graph of $y=\tan \theta$ ?

Sketch the graphs of $y=\tan \theta$ and $y=\sin \theta$ on the same grid for $-{180}^{\circ }<\theta <{180}^{\circ }$. How are the $\theta$-intercepts of the graph of $y=\sin \theta$ related to the graph of $y=\tan \theta$ ?