Deck 7: Trigonometry and Periodic Functions

Question 1

A vacationer sits all day on the corner of a pier in Boston Harbor and notices that at 9 am, when the water level is at its lowest, the water's depth is 2.5 feet. At $4 \mathrm{pm}$ , the water has risen to its maximum depth of 10.5 feet. If the depth of the water level varies periodically, let $f(t)$ be the formula for the depth of the water, in feet, as a function of time $t$ , in hours past 9 am. What is the period of the graph of $f(t)$ ?

Answer 1

14

Question 2

A ferris wheel is 34 meters in diameter, and must be boarded from a platform that is 1 meters above the ground. The wheel makes one complete revolution every 8 minutes. At the initial time $t=0$ , you are in the $12: 00$ position. If $h(t)$ gives your height above ground level $\mathrm{t}$ minutes after the initial time, what is the amplitude of $h(t)$ ?

Answer 2

17

Question 3

A ferris wheel is 49 meters in diameter, and must be boarded from a platform that is 1 meters above the ground. The wheel makes one complete revolution every 8 minutes. At the initial time $t=0$ , you are in the $12: 00$ position. If $h(t)$ gives your height above ground level $t$ minutes after the initial time, the midline of $h(t)$ is $y=$ ---------------.

Answer 3

25.5

Question 4

The graph below shows your height $h=f(t)$ in meters $t$ minutes after a ferris wheel ride begins. How many minutes are required for one complete revolution of the ferris wheel?

Question 5

The London ferris wheel is 135 meters in diameter and makes one revolution every 30 minutes. Let $y=h(t)$ be the height above ground after $t$ minutes of riding. Where is the midline for $h(t)$ ?

Question 6

The proposed London ferris wheel is 135 meters in diameter and makes one revolution every 30 minutes. Let $y=h(t)$ be the height above ground after $t$ minutes of riding. What does $h(2 t)$ represent?

A) The height on a wheel which is 2 times larger in diameter than the London one.
B) The height of a person who boarded the London ferris wheel 2 minutes before you.
C) The height on a ferris wheel that runs 2 times faster than the London one.
D) The height on the London ferris wheel with a loading platform 2 meters off the ground.

Question 7

Does the following function appear to be periodic with period $\leq 4$ ?

Question 8

Does the following function appear to be periodic with period less than or equal to 4 ?

Question 9

Estimate the period of the following periodic function.

A) $12 \pi$
B) $6 \pi$
C) -2
D) -4

Question 10

Estimate the period of the following periodic function.

Question 11

Suppose $f(a)=2 b$ and $f(2 a)=8 b$ . What is $f(3 a)$ if $f$ is periodic with period $2 a$ ? Your answer will have $b$ in it.

Question 12

Suppose $f(a)=5 b$ and $f(2 a)=9 b$ . What is $f(3 a)$ if $f$ is linear? Your answer will have $b$ in it.

Question 13

Suppose you are on a ferris wheel (that turns in a counter clockwise direction) and that your height, in meters, above the ground at time $t$ , in minutes, is given by $h(t)=16 \sin \left(\frac{\pi}{2} t\right)+18$ . How many meters above the ground are you at time $t=0 ?$

Question 14

Suppose you are on a ferris wheel (that turns in a counter clockwise direction) and that your height, in meters, above the ground at time $t$ , in minutes, is given by $h(t)=17 \sin \left(\frac{\pi}{2} t\right)+19$ . Your position on the wheel be at time $t=1$ is ---------oclock.

Question 15

Suppose you are on a ferris wheel (that turns in a counter clockwise direction) and that your height, in meters, above the ground at time $t$ , in minutes, is given by $h(t)=17 \sin \left(\frac{\pi}{2} t\right)+18$ . How many meters is the radius of the wheel?

Question 16

An animal population in a national park dropped from a high of 165,000 in 1943 to a low of 63,000 in 1989, and has risen since then. Scientists hypothesize that the population follows a sinusoidal cycle affected by predation and other environmental conditions, and that the caribou will again reach their previous high. Predict the next year when the population will again be 165,000 .

Question 17

Graph a function with midline 1 , amplitude 2, and period 4 . Show at least two full periods.

Question 18

Suppose the table below is for a periodic function $f$ with period 3:

What is the next integer value $n$ at which $f(n)=1$ ?

Question 19

Suppose the table below is for a periodic function $f$ with period 3:

Evaluate $f(86)$ .

Question 20

Which of the following graphs have period 5 and amplitude 2 ?

A)

B)

C)

D)

Question 21

The height above ground for a skydiver $t$ seconds after exiting the plane is given in the table.

Find the skydiver's average vertical speed between 6 and 8 seconds.

Question 22

A rocket is launched and its height above ground is given in the table.

Find the rocket's average vertical speed between 20 and 25 seconds.

Question 23

Given this graph of the distance of a pendulum from a wall,

what is the resting position of the pendulum (at the bottom of the swing) from the wall?

Question 24

Given this graph of the distance of a pendulum from a wall,

how far does the pendulum swing from the resting position?

Question 25

Find the point on the unit circle determined by the angle $140^{\circ}$ . (Round to three decimal places.)

Question 26

Find the angle that determines the point $(-0.766,0.643)$ on the unit circle. (Round to the nearest degree.)

Question 27

Find the reference angle for $343^{\circ}$ .

Question 28

Find the point on a circle with radius 4.5 determined by the angle $160^{\circ}$ . (Round to three decimal places.)

Question 29

What angle corresponds to 1.5 rotations around the unit circle?

Question 30

In the following figure, the coordinates of $Q$ are $(0.54,-0.84)$ . The angle $\theta=$ $\circ$ Round to the nearest whole degree.

Question 31

In the following figure, the coordinates of $Q$ are $(0.28,-0.96)$ . The coordinates of $\mathrm{P}$ are ( --------------,-----------). Round to 2 decimal places.

Question 32

In the following figure, the coordinates of $\mathrm{P}$ are $(0.41,0.91)$ . The angle $\theta=$ ----------° Round to the nearest whole number.

Question 33

In the following figure, the coordinates of $\mathrm{P}$ are $(0.47,0.88)$ . The angle $\phi=$ -------------° Round to the nearest whole number.

Question 34

In the following figure, the coordinates of $\mathrm{P}$ are $(0.37,0.93)$ . The coordinates of $\mathrm{R}$ are (-----------------,-------------- ). Round to the nearest whole number.

Question 35

In the following figure, the coordinates of $\mathrm{P}$ are $(0.36,0.93)$ . The angle $\theta=$ ------------- ° Round to the nearest degree.

Question 36

2.25 rotations around the unit circle corresponds to ------------°

Question 37

The coordinates of the point on the unit circle at the angle $330^{\circ}$ are $(\ldots, \ldots)$ . Round each coordinate to 3 decimal places.

Question 38

In the following figure, the circle shown is the unit circle. The coordinates of $Q$ are ( ---------------,-----------). Round to 2 decimal places.

Question 39

In the following figure, the circle shown is the unit circle. What is the horizontal distance from $Q$ to the $y$ -axis? Round to 2 decimal places.

Question 40

In the following figure, what is the length of segment $\overline{O A}$ ?

A) $\frac{5}{\cos \theta}$
B) $\frac{5}{\sin \theta}$
C) $5 \cos \theta$
D) $5 \sin \theta$

Question 41

A 16 foot ladder leans against the wall, forming a $65^{\circ}$ angle with the ground. How many feet from the wall is the foot of the ladder? Round to 2 decimal places.

Question 42

In order to measure its height, you stand 75 feet from a tree. The angle of sight is $30^{\circ}$ . How tall is the tree? Round your answer to the nearest hundredth of a foot.

Question 43

If the $x$ -value for the point on the unit circle with angle $\alpha^{\circ}$ is 0.39 , find $\cos \alpha$ and $\sin \alpha$ . Round your answers to three decimal places, if necessary.

Question 44

If the $y$ -value for the point on the unit circle with angle $\alpha^{\circ}$ is 0.39 , find $\cos \alpha$ and $\sin \alpha$ . Round your answers to three decimal places, if necessary.

Question 45

Find the coordinates of the point on the unit circle with angle $\alpha^{\circ}$ if $\cos \alpha=0.630$ . Round each coordinate to 3 decimal places.

Question 46

Find the coordinates of the point on the unit circle with angle $\alpha$ if $\sin \alpha=0.640$ . Round each coordinate to 3 decimal places.

Question 47

If angle $\alpha$ lies in Quadrant II, name the quadrant in which the following angles lie :

A) $90^{\circ}+\alpha$
B) $270^{\circ}-\alpha$
C) $720^{\circ}+\alpha$
D) $-180^{\circ}+\alpha$

Question 48

Consider the following figure. If $A=90^{\circ}, \sin C=0.49$ , and $c=7$ , find $a$ and $b$ . Round your answers to 2 decimal places, if necessary.

Question 49

Consider the following figure. If $A=90^{\circ}, \cos C=0.68$ , and $b=4$ , find $a$ and $c$ . Round your answers to 2 decimal places, if necessary.

Question 50

The coordinates of the point on a circle of radius 5 at the angle $240^{\circ}$ are (---------------,-------------). Round each coordinate to 3 decimal places.

Question 51

Find the coordinates of the point at angle $-25^{\circ}$ on a circle of radius 7.1. Round each coordinate to 3 decimal places, if necessary.

Question 52

What is $\sin ^{2} \theta$ for $\theta=\pi$ ?

Question 53

The following figure shows the path taken by the left front tire of a car as it changes direction sharply from due north to due east. The turning radius of the car is 20 feet, and the two front wheels are 56 inches apart. How many feet does the left front wheel travel from $A$ to $B$ ? Round to 1 decimal place.

Question 54

The following figure shows the path taken by the left front tire of a car as it changes direction sharply from due north to due east. The turning radius of the car is 18 feet, and the two front wheels are 57 inches apart. While the left front wheel travels from $A$ to $B$ , how many feet does the right front wheel travel? Round to 1 decimal place.

Question 55

The minute hand on a watch is 0.5 inches long. How many inches does the tip of the minute hand travel as the hand turns through $219^{\circ}$ ? Round to 2 decimal places.

Question 56

The minute hand on a clock is 3.6 inches long. How many inches per minute does the tip of the minute hand travel? Round to 3 decimal places.

Question 57

In the revolving door in the figure below, each panel is 1.1 meters long. How many meters is the arc length between $A$ and $D$ ? Round to 2 decimal places.

Question 58

Without a calculator, what is the exact value of $\cos \left(\frac{2 \pi}{3}\right)$ ?

Question 59

What is the reference angle for $-\frac{7 \pi}{4}$ ?

A) 0
B) $\frac{\pi}{4}$
C) $\frac{\pi}{6}$
D) $\frac{\pi}{3}$

Question 60

The arc length corresponding to $90^{\circ}$ on a circle of radius 2.5 is $\pi$ ------------- . Round to 2 decimal places.

Question 61

The angle $315^{\circ}$ is equivalent to ------------ $\pi$ radians. Round your answer to 2 decimal places.

Question 62

The angle $\frac{3 \pi}{4}$ radians is equivalent to --------------°

Question 63

0.5 rotations around the unit circle corresponds to -------------- $\pi$ radians.

Question 64

What is the length of an arc cut off by an angle of $210^{\circ}$ in a circle of radius 2.5 meters? Give your answer correct to 3 decimal places.

Question 65

An ant starts at the point $(-1,0)$ and walks 1.5 units around the unit circle in a clockwise direction. Find the $x$ and $y$ coordinates (accurate to 2 decimal places) of the final location of the ant.

Question 66

Find the sign of the following:
a) $\sin (7 \mathrm{rad})$
b) $\cos (7 \mathrm{rad})$

Question 67

The number $(\sin (10 \mathrm{rad}))(\cos (7 \mathrm{rad}))$ is positive.

Question 68

A wheel has radius 6 inches and spins at a rate of 11 revolutions per second. A red dot is painted on the outermost part of the wheel. How far (in inches) has the dot traveled in 7 seconds? Round your answer to 3 decimal place.

Question 69

A wheel has radius 6 inches and spins at a rate of 766 degrees per second. A red dot is painted on the outermost part of the wheel. How far (inches) has the dot traveled in 7 seconds? Round your answer to 3 decimal places.

Question 70

Which is larger: $\sin \left(\frac{3 \pi}{7}\right)$ or $\cos \left(\frac{3 \pi}{7}\right)$ ?

Question 71

Which is larger: $\sin \left(\frac{9 \pi}{7}\right)$ or $\cos \left(\frac{9 \pi}{7}\right)$ ?

Question 72

The coordinates of the point on a circle of radius 5 at the angle $240^{\circ}$ are ( ------------,------------). Round each coordinate to 3 decimal places.

Question 73

Without a calculator, find the exact value of $\cos 180^{\circ}$ .

Question 74

Find the coordinates of the point at angle $-26^{\circ}$ on a circle of radius 7.1. Round each coordinate to 3 decimal places, if necessary.

Question 75

A circle of radius 3 is centered at the origin. Point A lies on the circle at angle $201^{\circ}$ . Point $\mathrm{B}$ lies outside the circle and has coordinates $(5,-2)$ . What is the distance between these two points? Round your answer to 2 decimal places.

Question 76

What angle in radians corresponds to 2.3 rotations around the unit circle? Round to 2 decimal places.

Question 77

Find the arc length corresponding to an angle of $\frac{\pi}{4}$ radians on a circle of radius 3.9.
Round to 2 decimal places.

Question 78

The point $\mathrm{Q}$ in the following figure is at the lowest point on the sine-curve. As $k$ increases, does the value of $m$ increase, decrease, or stay the same?

Question 79

City $A$ is a modest tourist town, which means that its population undergoes a seasonal variation. In January, it dips down to 4,000 people. By July, with the warm weather, its population climbs to around 5,000 people. This trend repeats every year. City $B$ , on the other hand, is a small town not far from City $A$ . Its population has been growing extremely rapidly ever since the arrival of several large industrial plants. There were only 3,500 people living in City $B$ on January 1, 2004, but its population has been growing by $10 \%$ every year thereafter. At how many different points in time will the population of City $A$ equal the population of City $B$ ?

Question 80

City $A$ is a modest tourist town, which means that its population undergoes a seasonal variation. In January, it dips down to 6000 people. By July, with the warm weather, its population climbs to around 8000 people. This trend repeats every year. City $B$ , on the other hand, is a small town not far from City $A$ . Its population has been growing extremely rapidly ever since the arrival of several large industrial plants. There were only 5500 people living in City $B$ on January 1, 2000, but its population has been growing by $9 \%$ every year thereafter. During which year did population of City $A$ last equal the population of City $B$ ?