Deck 9: Trigonometric Identities and Their Applications

Question 1

The following table gives , the percentage of the electorate favoring candidate A during the 12 months preceding a presidential election. Time, t, is measured in months, and t = 0 is a year before election day. If were approximately trigonometric, its formula could be written . Round the second answer to 3 decimal places.

Answer 1

a)-18
b)0.167
c)45

Question 2

The following table gives , the percentage of the electorate favoring candidate A during the 12 months preceding a presidential election. Time, t, is measured in months, and t = 0 is a year before election day. Assume that is approximately trigonometric. A second candidate, candidate B, has a percentage of support given by . What is the largest value of t, , at which the two candidates are tied for electoral support? Round to 2 decimal places.

Answer 2

11.39

Question 3

The following table gives, the percentage of the electorate favoring candidate A during the 12 months preceding a presidential election. Time, t, is measured in months, and t = 0 is a year before election day.
Assume thatIs approximately trigonometric. A second candidate, candidate B, has a percentage of candidate support given by LetFor

What is the meaning of the maximum of?

A) The maximum percentage lead candidate A has over candidate B.
B) The maximum percentage lead candidate B has over candidate A .
C) The maximum combined percentage of the electorate favoring either candidate A or candidate B.
D) The maximum combined percentage of the electorate favoring neither candidate A nor candidate B.

Answer 3

The maximum combined percentage of the electorate favoring either candidate A or candidate B.

Question 4

Two weights (weight 1 and weight 2) are suspended from the ceiling by springs. At time t = 0 (t in seconds), the weights are set in motion and begin bobbing up and down. Eventually, however, the oscillation of both weights dies down. The following equations describe the distance of each weight from the ceiling as a function of time:
and .

Which weight is closer to the ceiling at time t = 2?

A) Weight 1
B) Weight 2

Question 5

Two weights (weight 1 and weight 2) are suspended from the ceiling by springs. At time t = 0 (t in seconds), the weights are set in motion and begin bobbing up and down. Eventually, however, the oscillation of both weights dies down. The following equations describe the distance of each weight from the ceiling as a function of time:
and .

Which weight has oscillations which die down the fastest?

A) Weight 2
B) Weight 1

Question 6

Two weights (weight 1 and weight 2) are suspended from the ceiling by springs. At time t = 0 (t in seconds), the weights are set in motion and begin bobbing up and down. Eventually, however, the oscillation of both weights dies down. The following equations describe the distance of each weight from the ceiling as a function of time:
and .
At what time are the two weights farthest apart?

A) At t = 0.
B) Between t = 0 and t = 0.5.
C) At t = 0.5.
D) Between t = 0.5 and t = 1.
E) At t = 1.

Question 7

Is the square of a complex number always real and nonnegative?

Question 8

Find a formula for a deer population which oscillates over a 6 year period between a low of 1000 in year t=0 and a high of 2900 in year t=3 .

Question 9

The deer population in a state park is modelled by where t is the number of months since January 1, 2005. Evaluate

and interpret the result. Round to the nearest whole number.

Question 10

The deer population in a state park is modelled by where t is the number of months since January 1, 2005. If , find the value(s) of t at which the deer population is equal to 280. Round your answer to the nearest tenth.

Question 11

A ferris wheel sitting on the ground is 24 meters in diameter and makes one revolution every 7 minutes. If you start in the 9 o'clock position t= 0 and the wheel is rotating clockwise, write a formula for your height above the ground at time t.

Question 12

A ferris wheel sitting on the ground is 26 meters in diameter and makes one revolution every 7 minutes. If you start in the 9 o'clock position t= 0 and the wheel is rotating counterclockwise, write a formula for your height above the ground at time t.

Question 13

A ferris wheel sitting on the ground is 20 meters in diameter and makes one revolution every 7 minutes. If you start in the 9 o'clock position t= 0 and the wheel is rotating counterclockwise, write a formula for your height above the ground at time t.

Question 14

A ferris wheel sitting on the ground is 22 meters in diameter and makes one revolution every 5 minutes. If you start in the 9 o'clock position at t= 0 and the wheel is rotating clockwise, when is the first time that are you 16.5 meters above the ground?

Question 15

A ferris wheel sitting on the ground is 24 meters in diameter and makes one revolution every 7 minutes. If you start in the 9 o'clock position at t= 0 and the wheel is rotating counterclockwise, when is the first time that are you 6 meters above the ground?

Question 16

A mass attached to a spring moves horizontally on a frictionless track. Its displacement from the rest position at time t is given by . What is the furthest distance from the rest position that the mass will achieve? The displacement is measured in meters.

Question 17

A mass attached to a spring moves horizontally on a frictionless track. Its displacement from the rest position at time t is given by . If displacement is measured in inches and time is measured in inches, when is the mass 0.3 inches from the rest position? Restrict your answer(s) to , and round to 3 decimal places.

Question 18

A mass attached to a spring moves horizontally on a frictionless track. Its velocity at time t is given by . What is the maximum velocity that the mass will achieve? The displacement is measured in meters and the time is measured in seconds.

Question 19

The population in a town oscillates over a 15 year period beginning with a high of 3000 people in year and a low of 2300. Find a formula for the town's population.

Question 20

The population in a town oscillates over a 7 year period beginning with a high of 3000 people in year t= 0 and a low of 2300. Find a formula for the town's population.

A)
B)
C)
D)

Question 21

Does ?

Question 22

If can also be written in the form .

Question 23

Does

Question 24

If ,can also be written in the form ?

Question 25

Does ?

Question 26

Using the sum or difference formulas, . Round both answers to 4 decimal places.

Question 27

Using the sum or difference formulas, . Round all answers to 4 decimal places.

Question 28

Does ?

Question 29

Does ?

Question 30

Find the exact value of .

Question 31

Find the exact value of .

Question 32

Find the smallest value of t such that and .

Question 33

Find the smallest value of t such that and .

Question 34

Find the smallest value of t such that and

Question 35

Find the smallest value of t such that and .

Question 36

Write in the form .

Question 37

Write in the form
.

Question 38

Write in the form .

Question 39

Calculate exactly.

Question 40

Write In the form
.

A)
B)
C)
D)

Question 41

Does ?

Question 42

What is the smallest positive solution to ? Round to 2 decimal places.

Question 43

What is the smallest positive solution to ? Round to 2 decimal places.

Question 44

How many solutions does have for ?

Question 45

What is the smallest positive solution to ? Round to 2 decimal places.

Question 46

How many solutions does have for ?

Question 47

Does ?

Question 48

What is the smallest positive solution to ? Round your answer to 2 decimal places.

Question 49

How many solutions does have for ?

Question 50

Which of the following statements are identities?

A)
B)
C)
D)
E)
F)

Question 51

What is for ?

Question 52

Either show the following equation is true, or find a value of x for which the equation is false:

Question 53

Either show the following equation is true, or find a value of x for which the equation is false:

Question 54

Either show the following equation is true, or find a value of x for which the equation is false:

Question 55

Write in terms of the tangent function.

Question 56

Write in terms of the cotangent function.

Question 57

Question 58

Question 59

How many solutions does have for ?

A)4
B)0
C)1
D) none of the above.

Question 60

Write in terms of the tangent function.

A)
B)
C)
D)