Deck 9: Trigonometric Identities, Models, and Complex Numbers

Question 1

At which of the following values of $x$ do the graphs of $y=25+\frac{1}{2} t$ and $y=25+\frac{1}{2} t-2 \cos \left(\frac{t}{2}\right)$ intersect?

A) All multiples of $\pi$
B) All even multiples of $\pi$
C) All odd multiples of $\pi$
D) None of the above

Answer 1

All odd multiples of $\pi$

Question 2

Estimate the solution to the equation $\sin t=0.7$ for $0 \leq t \leq \pi / 2$ . Round to 2 decimal places.

Answer 2

0.78

Question 3

Find a solution in the interval $-\frac{\pi}{2}<\theta<\frac{\pi}{2}$ for $\tan \theta=-5$ . Round to 3 decimal places.

Answer 3

$-1.373 \mathrm{rad}$

Question 4

Use a graph of $y=\sin t$ to estimate the solution to the equation $\sin t=0.2$ for $\pi / 2 \leq t \leq \pi$ . Round to 2 decimal places.

Question 5

Which of the following are $x$ -intercepts of the function $f(x)=-\sin \left(\frac{x}{2}\right)-4$ ?

A) 0
B) $\pi$
C) $2 \pi$
D) $3 \pi$
E) $4 \pi$
F) $6 \pi$
G) None of the above

Question 6

Find all solutions to $\sin \theta=0.8$ on the interval $3 \pi \leq \theta \leq 5 \pi$ . Give your answers correct to 3 decimal places.

Question 7

Find all solutions to the equation $7 \sin (x-1)=2$ for $0 \leq x \leq 2 \pi$ . Give your answers to 3 decimal places.

Question 8

The baseball field's usage (in people per week) is seasonal with the peak in mid-July and the low in mid-January. The usage is 2,000 in July and 500 in January. Find a trig function $s=f(t)$ representing the usage at time $t$ months after mid-January.

Question 9

Which of the following are defined?

A) $\sin ^{-1} 3.45$
B) $\cos ^{-1} 3.45$
C) $\tan ^{-1} 3.45$
D) $\sec ^{-1} 3.45$
E) $\csc ^{-1} 3.45$
F) $\cot ^{-1} 3.45$

Question 10

Find all values of $\theta$ such that $\tan \theta=3.161$ and $360^{\circ} \leq \theta \leq 720^{\circ}$ . Give answers correct to 3 decimal places.

Question 11

Find all solutions (if possible) for $t$ in radians: $4 \sin t=3$ . Give answers correct to 3 decimal places.

Question 12

Find all solutions (if possible) with $t$ in radians: $2 \cos (t-1)=4$ . Give answers correct to 3 decimal places.

Question 13

Find all solutions to $2 \cos t=0.14$ for $-2 \pi<t<-\pi$ . Give answers correct to 3 decimal places.

Question 14

Find all solutions to $-3 \cos t-\sin t \cos t=0$ for $0 \leq t \leq 2 \pi$ . Give answers correct to 3 decimal places.

Question 15

How many solutions to $\cos x=\frac{1}{4}$ are there for $0 \leq x \leq \pi$ ?

Question 16

How many solutions to $\sin x=\frac{1}{6}$ are there for $0 \leq x \leq \pi$ ?

Question 17

How many solutions to $\sin x=\frac{-1}{3}$ are there for $0 \leq x \leq \pi$ ?

Question 18

Solve $\tan x-2 \sin x=0$ for $0 \leq x \leq 2 \pi$ .

Question 19

Solve $\csc x=2$ for $0 \leq x \leq 2 \pi$ .

Question 20

Solve $\tan x=3$ for $0 \leq x \leq 2 \pi$ .

Question 21

Graph $y=\sin t$ and use that graph to approximate the solution of $0.65=\sin t$ on $0 \leq t \leq \frac{\pi}{2}$ . Give $t$ in degrees to 2 decimal places.

Question 22

Graph $y=\cos t$ and use that graph to approximate the solution of $0.12=\cos t$ on $0 \leq t \leq \frac{\pi}{2}$ . Give $t$ in degrees to 2 decimal places.

Question 23

Graph $y=\tan t$ and use that graph to approximate the solution of $0.49=\tan t$ on $0 \leq t \leq \frac{\pi}{2}$ . Give $t$ in degrees to 2 decimal places.

Question 24

Does $\sin \frac{1}{2} t=\sqrt{\frac{1-\cos t}{2}}$ ?

Question 25

What is the smallest positive solution to $2 \sin \theta\left(\cos \theta+\frac{1}{\sin \theta}\right)=2.5$ ? Round to 2 decimal places.

Question 26

What is the smallest positive solution to $5 \sin ^{2} x-\cos ^{2} x=1$ ? Round to 2 decimal places.

Question 27

How many solutions does $4 \sin ^{2} x-\cos ^{2} x=4$ have for $0 \leq x \leq 2 \pi$ ?

Question 28

What is the smallest positive solution to $\cos ^{3} x+\cos x \sin ^{2} x=4 \sin x$ ? Round to 2 decimal places.

Question 29

How many solutions does $\cos ^{3} x+\cos x \sin ^{2} x=4 \sin x$ have for $-\pi \leq x \leq 2 \pi$ ?

Question 30

Does $\frac{(\cos \theta-\sin \theta)^{2}-(\cos \theta+\sin \theta)^{2}}{2 \sin 2 \theta}=1$ ?

Question 31

What is the smallest positive solution to $\cos ^{4} \theta-\sin ^{4} \theta=\frac{1}{4}$ ? Round your answer to 2 decimal places.

Question 32

How many solutions does $\cos ^{4} \theta-\sin ^{4} \theta=\frac{1}{3}$ have for $0 \leq \theta \leq 4 \pi$ ?

Question 33

Which of the following statements are identities?

A) $\sin \theta=\cos \theta$
B) $\sin (x)=\sin (x+4 \pi)$
C) $\sin ^{2} t+\cos ^{2} t=1$
D) $\cos (2 \theta)=2 \cos \theta \sin \theta$
E) $\csc x=\frac{1}{\sin x}$
F) $\cos \theta=\frac{\sqrt{3}}{2}$

Question 34

What is $\sin 2 \theta$ for $\theta=\pi / 4$ ?

Question 35

Write $\frac{5 \sin (x-5)}{8 \cos (x-5)}$ in terms of the tangent function.

Question 36

Write $\frac{3 \cos (x-6)}{8 \sin (x-6)}$ in terms of the cotangent function.

Question 37

If $\pi<\theta<\frac{3 \pi}{2}$ and $\cos (\theta)=\frac{4}{11}$ , find $\cos (2 \theta), \sin (2 \theta)$ , and $\tan (2 \theta)$ exactly.

Question 38

If $\frac{3 \pi}{2}<\theta<2 \pi$ and $\sin (\theta)=\frac{-8}{9}$ , find $\sin (2 \theta), \cos (2 \theta)$ , and $\tan (2 \theta)$ exactly.

Question 39

Write $\frac{3 \sin (x-3)}{4 \cos (x-3)}$ in terms of the tangent function.

A) $\frac{3}{4} \tan (x-3)$
B) $3 \tan (x-3)$
C) $\frac{3}{4} \cdot \frac{\tan (x-3)}{x-3}$
D) $\frac{3}{4 \tan (x-3)}$

Question 40

How many solutions does $6 \sin ^{2} x-\cos ^{2} x=3$ have for $0 \leq x \leq 2 \pi$ ?

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

Question 41

If $0 \leq t \leq \pi$ , in what quadrant is $\cos t=-0.69$ ?

A) I
B) II
C) III
D) IV

Question 42

If $-\frac{\pi}{2} \leq t \leq \frac{\pi}{2}$ , in what quadrant is $\sin t=-0.64$ ?

A) I
B) II
C) III
D) IV

Question 43

If $0 \leq t \leq \pi$ , in what quadrant is $\cos t=0.31$ ?

A) I
B) II
C) III
D) IV

Question 44

If $-\frac{\pi}{2} \leq t \leq \frac{\pi}{2}$ , in what quadrant is $\sin t=0.61$ ?

A) I
B) II
C) III
D) IV

Question 45

Either show the following equation is true, or find a value of $x$ for which the equation is false:
$\sin (4 x)+\sin (2 x)=\sin (6 x)$

Question 46

Either show the following equation is true, or find a value of $x$ for which the equation is false:
$\cos (3 x)+\cos (2 x)=\cos (5 x)$

Question 47

Either show the following equation is true, or find a value of $x$ for which the equation is false:
$\sin (8)=2 \sin (4) \cos (4)$

Question 48

Does $\sin u+\sin v=2 \sin \left(\frac{u-v}{2}\right) \cos \left(\frac{u+v}{2}\right)$ ?

Question 49

If $s(x)=\sin (3 x)+\sin (4 x)$ , then $s(x)$ can also be written in the form
$s(x)=$ ---------- $\sin ($ -------------- $x)(\cos$ ---------- $x)$ .

Question 50

Does $\cos (3 t)=3 \cos ^{3} t-4 \cos t$ ?

Question 51

Does $\cos \left(-75^{\circ}\right)=\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}+\frac{1}{2} \cdot \frac{\sqrt{2}}{2} ?$

Question 52

Using the sum or difference formulas, $3 \sin t+2 \cos t=\ldots \sin (t+\ldots)$ .
Round both answers to 4 decimal places.

Question 53

Using the sum or difference formulas,
$3 \sin 2 t-4 \cos 2 t=\ldots \sin (\ldots t+\ldots)$ . Round all answers to 4 decimal places if necessary.

Question 54

Does $\sin 105^{\circ}=\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}+\frac{1}{2} \cdot \frac{\sqrt{2}}{2}$ ?

Question 55

Find the exact value of $\cos 435^{\circ}$ .

Question 56

Find the exact value of $\sin 525^{\circ}$ .

Question 57

Find the smallest value of $t$ such that $t>0$ and $\cos (10 t)-\cos (5 t)=0$ .

Question 58

Find the smallest value of $t$ such that $t>0$ and $\cos (10 t)+\cos (9 t)=0$ .

Question 59

Find the smallest value of $t$ such that $t>0$ and $\sin (8 t)-\sin (3 t)=0$ .

Question 60

Find the smallest value of $t$ such that $t>0$ and $\sin (4 t)+\sin (5 t)=0$ .

Question 61

Calculate $\tan 15^{\circ}$ exactly.

Question 62

Does $\cos 345^{\circ}=\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}-\frac{1}{2} \cdot \frac{\sqrt{2}}{2}$ ?

Question 63

Show that $\cos 3 t=4 \cos ^{3} t-3 \cos t$ .

Question 64

Show that $\sin 3 t=3 \sin t-4 \sin ^{3} t$ .

Question 65

Does $\cos (3 t)=3 \cos ^{3} t-3 \cos t$ ?

Question 66

If $f(x)=5 \sin (8 \pi x)$ , can $f(x)$ also be written in the form $f(x)=5 \cos \left(8 \pi x-\frac{\pi}{2}\right)$ ?

Question 67

Using the sum or difference formulas, $5 \sin t+3 \cos t=$ --------- $\sin (t+$ -----------). Round both answers to 4 decimal places.

Question 68

Using the sum or difference formulas,
$2 \sin 3 t-4 \cos 3 t=$ ------------- $\sin ( ---------- t+$ ------------). Round all answers to 4 decimal places if necessary.

Question 69

Write $10 \sin (4 t)-9 \cos (4 t)$ in the form $A \sin (B t+C)$ . Round all numbers to 3 decimal places if necessary.

Question 70

Write $-8 \sin (3 t)+11 \cos (3 t)$ in the form $A \sin (B t+C)$ . Round all numbers to 3 decimal places if necessary.

Question 71

Write $-6 \sin (3 t)-7 \cos (3 t)$ in the form $A \sin (B t+C)$ . Round all numbers to 3 decimal places if necessary.

Question 72

Write $8 \sin (4 t)-9 \cos (4 t)$ in the form $A \sin (B t+C)$ . Round all numbers to 3 decimal places if necessary.

A) $12.042 \sin (4 t-0.844)$
B) $8 \sin (4 t-0.844)$
C) $12.042 \sin (4 t+8.5)$
D) $12.042 \sin (-0.844 t+4)$

Question 73

The following table gives $A(t)$ , 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 $A(t)$ were approximately trigonometric, its formula could be written $A(t)=----------\cos (\ldots \pi t)+\ldots$ . Round the second answer to 3 decimal places.

Question 74

The following table gives $A(t)$ , 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 $A(t)$ is approximately trigonometric. A second candidate, candidate $B$ , has a percentage of support given by $B(t)=30+15 \sin \left(\frac{\pi}{6} t\right)$ . What is the largest value of $t, 0 \leq t \leq 12$ , at which the two candidates are tied for electoral support? Round to 2 decimal places.

Question 75

The following table gives $A(t)$ , 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 $A(t)$ is approximately trigonometric. A second candidate, candidate $B$ , has a percentage of candidate support given by $B(t)=31+15 \sin \left(\frac{\pi}{6} t\right)$ . Let $f(t)=A(t)+B(t)$ for $0 \leq t \leq 12$ . What is the meaning of the minimum of $f(t)$ ?

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

Question 76

Two weights (weight 1 and weight 2 ) are suspended from the ceiling by springs. At time $\mathrm{t}=0$ ( $\mathrm{t}$ 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:
$d_{1}=4+3 \cos (\pi t) e^{-0.3 t} \text { and } d_{2}=2+2 \cos (2 \pi t) e^{-0.5 t} \text {. }$
Which weight is closer to the ceiling at time $t=20$ ?

A) Weight 1
B) Weight 2

Question 77

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:
$d_{1}=5+4 \cos (\pi t) e^{-0.3 t} \text { and } d_{2}=3+3 \cos (2 \pi t) e^{-0.5 t}$
Which weight has oscillations which die down the slowest?

A) Weight 1
B) Weight 2

Question 78

Two weights (weight 1 and weight 2 ) are suspended from the ceiling by springs. At time $\mathrm{t}=0$ ( $\mathrm{t}$ 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:
$d_{1}=6+5 \cos (\pi t) e^{-0.1 t} \text { and } d_{2}=5+4 \cos (2 \pi t) e^{-0.3 t}$
At what time are the two weights furthest 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 79

Find a formula for a deer population which oscillates over a 6 year period between a low of 1,000 in year $t=0$ and a high of 2,700 in year $t=3$ .

Question 80

The deer population in a state park is modelled by $f(t)=55 \sin \left(\frac{\pi t}{6}\right)+220$ where $t$ is the number of months since January 1, 2005. Evaluate $f(6)-f(3)$ and interpret the result. Round to the nearest whole number.