Deck 7: Conic Sections

Question 1

Suppose that the terminal point determined by t is the point $\left( \frac { 5 } { 13 } , \frac { 12 } { 13 } \right)$ on the unit circle. Find the terminal point determined by $\pi + t$ .

Answer 1

$\left( - \frac { 5 } { 15 } , - \frac { 12 } { 13 } \right)$

Question 2

Find the sign of $\cos t \sin t$ if the terminal point determined by t is in quadrant II.

Answer 2

negative

Question 3

Find the exact value of $\text { Sec } \left( \frac { 3 \pi } { 4 } \right)$ and $\operatorname { CSC } \left( - \frac { 11 \pi } { 4 } \right)$ .

Answer 3

$- \sqrt { 2 }$ , $- \sqrt { 2 }$

Question 4

Find the period of the function $y = 3 \sec x$ and sketch its graph.

Question 5

Find the period of the function $y = ( \cot x ) / 5$ and sketch its graph.

Question 6

Find period and graph the function. $y = 2 \tan ( 2 x - \pi / 4 )$

Question 7

Find the reference number and the terminal point $p( x , y )$ determined by $t = \frac { 11 \pi } { 6 }$ .

Question 8

The point $P ( x , y )$ is on the unit circle in quadrant IV. If $y = - 5 / 6$ find x.

Question 9

Find to two decimal places the approximate value of

Question 10

Use the fundamental identities to write the first expression in terms of the second.

Question 11

If the terminal point determined by t is $\left(\frac{12}{15},-\frac{5}{15}\right)$ , find $\sin t$ , $\cos t$ and $\tan t$ .

Question 12

Find the vertical asymptotes for the function $y = 3 \sec \frac { 1 } { 2 } x$ in the interval $[ 0,4 \pi ]$ .

Question 13

If $605 t = 3 / 5$ and the terminal point for t is in quadrant IV, find $\tan t + \mathrm { sec } t$ .

Question 14

Find the reference number for
.

Question 15

Find $\tan t$ given that $\sin t = - \frac { 3 } { 5 }$ and $\mathrm { cot } t < 0$ .

Question 16

Find the amplitude, period and phase shift of
.

Question 17

Find the exact value of $\operatorname { sit } \left( \frac { 9 x } { 2 } \right)$ and $\operatorname { Cos} \left( - \frac { 9 x } { 2 } \right)$ .

Question 18

Use a graphing device to find the maximum and minimum values of the function $y = \cos x - \frac { 1 } { 2 } \cos 2 x$ .

Question 19

Show that the point $( 1 / \sqrt { 5 } , - 2 / \sqrt { 5 } )$ is on the unit circle.

Question 20

Find the amplitude, period, and phase shift of the function. $y = 2 \cos \left( \frac { 1 } { 2 } x - \frac { \pi } { 6 } \right)$

Question 21

Suppose that the terminal point determined by t is the point $\left( \frac { 5 } { 13 } , \frac { 12 } { 13 } \right)$ on the unit circle. Find the terminal point determined by $2 \pi - t$ .

Question 22

The point
is on the unit circle in quadrant II. If
find y.

Question 23

The function $y = - 2 \cos \left( \frac { t } { 3 } \right)$ models the displacement of an object moving in simple harmonic motion, where y is measured in inches and t in seconds. Find the amplitude, period, and frequency of motion and sketch a graph of the function over one complete period.

Question 24

Find the period of the function
and sketch its graph.

Question 25

Show that the point $\left( - \frac { 2 \sqrt { 2 } } { 3 } , - \frac { 1 } { 3 } \right)$ is on the unit circle.

Question 26

If the terminal point determined by t is $\left( \frac { 12 } { 13} , - \frac { 5 } { 13 } \right)$ , find $\sin t$ , $\cos t$ and $\tan t$ .

Question 27

Find the exact value of the expression, if it is defined.

Question 28

Find the exact value of
and
.

Question 29

Determine whether the function is even, odd, or neither.

Question 30

Find the reference number for
.

Question 31

Find a function that models simple harmonic motion having the given properties. Assume that the displacement is zero at time
.amplitude 10 in, frequency

Question 32

Find $\operatorname { sect }$ given that $\sin t = - \frac { 3 } { 5 }$ and $\tan t > 0$ .

Question 33

If $\cos t = \frac { 3 } { 5 }$ and the terminal point for t is in quadrant IV, find $\tan t + \mathrm { sec } t$ .

Question 34

Find to two decimal places the value of
using a calculator.

Question 35

Find the reference number and the terminal point $p( x , y )$ determined by $t = \frac { 19 \pi } { 6 }$ .

Question 36

Express $\tan t$ in terms of $\sin t$ , if the terminal point determined by t is in quadrant III.

Question 37

Find the exact value of $\sin \left( \frac { 19 \pi } { 6 } \right)$ and $\mathrm { cos } \left( - \frac { 5 \pi } { 6 } \right)$ .

Question 38

Find period and graph the function. $y = 2 \tan ( x - \pi / 4 )$

Question 39

Determine whether the function is even, odd, or neither.

Question 40

If the terminal point determined by t is $\left( \frac { 12 } { 13 } , - \frac { 5 } { 13 } \right)$ , find $\sin t$ , $\cos t$ And $\tan t$

A) $\sin ( t ) = \frac { 12 } { 13 }$ , $\cos \left( t ^ { \prime } \right) = \frac { 5 } { 13 }$ , $\tan ( t ) = \frac { 5 } { 12 }$

B) $\sin ( t ) = - \frac { 5 } { 13 }$ , $\cos (t)=\frac{12}{13}$ , $\tan ( t ) = - \frac { 5 } { 12 }$

C) $\sin ( t ) = \frac { 5 } { 13 }$ , $\cos ( t ) = - \frac { 12 } { 13 }$ , $\tan ( t ) = - \frac { 13 } { 12 }$

D) $\sin ( t ) = - \frac { 5 } { 13 }$ , $\cos = \frac { 12 } { 13 }$ , $\tan ( t ) = - \frac { 12 } { 5 }$

E) $\sin ( t ) = - \frac { 5 } { 13 }$ , $\cos = \frac { 12 } { 13 }$ , $\text { tant } t = \frac { 12 } { 5 }$

Question 41

Find $\tan t$ given that $\sin t = \frac { 3 } { 5 }$ and $\mathrm { cot } t < 0$

A) $\tan t = - \frac { 3 } { 4 }$

B) $\tan t = - \frac { 3 } { 5 }$

C) $\tan t = - 1$

D) $\tan t = \frac { 3 } { 4 }$

E) $\tan t = - \frac { \sqrt { 3 } } { 2 }$

Question 42

Suppose that the terminal point determined by t is the point $\left( \frac { 3 } { 5 } , - \frac { 4 } { 5 } \right)$ on the unit circle. Find the terminal point determined by $\pi + t$

A) $\left( \frac { 3 } { 3 } , - \frac { 4 } { 5 } \right)$

B) $\left( - \frac { 3 } { 5 } , \frac { 4 } { 5 } \right)$

C) $\left( \frac { 3 } { 4 } , - \frac { 4 } { 3 } \right)$

D) $( - 1,1 )$

E) $\left( - \frac { 4 } { 5 } , \frac { 3 } { 5 } \right)$

Question 43

If $\cos t = \frac { 3 } { 5 }$ and the terminal point for t is in quadrant IV, find $\mathrm { Cot } t + \mathrm { CSC } t$

A) $- \frac { 1 } { 2 }$

B) $- 2$

C) $2$

D) $\frac { 1 } { 2 }$

E) $- \frac { 5 } { 4 }$

Question 44

Find the vertical asymptotes for the function $y = \tan 2 x$ in the interval $\left( - \frac { \pi } { 2 } , \frac { \pi} { 2 } \right\}$ .

Question 45

Find the exact value of $\text { Sec } \left( \frac { 3 \pi } { 4 } \right)$ and $\operatorname { CSC} \left( - \frac { 11 \pi } { 4 } \right)$

A) $\frac { - \sqrt { 2 } } { 2 }$ , $- \sqrt { 2 }$

B) $- \sqrt { 2 }$ , $\frac { - 1 } { 2 }$

C) $\frac { - \sqrt { 2 } } { 3 }$ , $- \frac { \sqrt { 2 } } { 3 }$

D) $- \sqrt { 2 }$ , $- \sqrt { 2 }$

E) $- 1$ , $- 1$

Question 46

Find the exact value of each expression, if it is defined
a) $\sin ^ { - 1 } ( - 2 )$

b) $\cos ^ { - 1 } ( - \sqrt { 3 } )$ c) $\tan ^ { - 1 } ( - 1 )$

Question 47

Find the reference number for $t = - \frac { 26 \pi } { 5 }$

A) $- \frac { \pi } { 5 }$

B) $\frac { 26 \pi } { 5 }$

C) $- \frac { 13 \pi } { 5 }$

D) $5 \pi$

E) $\frac { \pi } { 5 }$

Question 48

Find a function that models simple harmonic motion having the given properties. Assume that the displacement is maximum at time $t = 0$ .amplitude $2.5 \mathrm {~m}$ , frequency $750 \mathrm {~Hz}$

Question 49

Find the exact value of $\sin \left( \frac { 5 \pi } { 2 } \right)$ and $\text {cos} \left( - \frac { 5 \pi } { 2 } \right)$

A) $\sin \left( \frac { 5 \pi } { 2 } \right) = \frac { 5 } { 2 }$ , $\mathrm { cos } \left( - \frac { 5 \pi } { 2 } \right) = \frac { 3 } { 2 }$

B) $\sin \left( \frac { 5 \pi } { 2 } \right) = - 1$ , $\cos \left( - \frac { 5 \pi } { 2 } \right) = 0$

C) $\sin \left( \frac { 5 \pi } { 2 } \right) = 0$ , $\cos \left( - \frac { 5 \pi } { 2 } \right) = - 1$

D) $\sin \left( \frac { 5 \pi } { 2 } \right) = 1$ , $\cos \left( - \frac { 5 \pi } { 2 } \right) = 0$

E) $\sin \left( \frac { 5 \pi } { 2 } \right) = - 1$ , $cos \left( - \frac { 5 \pi } { 2 } \right) = 1$

Question 50

Use a graphing device to find the maximum and minimum values of the function
.

Question 51

Find the sign of $\cos t \sin t$ and $\text { sect }$ If the terminal point determined by tIs in quadrant II

A) $+ , -$
B) $- , +$
C) $- , -$
D) $+ , +$
E) $+ \text {, undefined }$

Question 52

The point is on the unit circle in quadrant III. If
Find y

A)
B)
C)
D)
E)

Question 53

Find the amplitude, period, and phase shift of the function.

Question 54

Suppose that point $P \left( \frac { \sqrt { 5 } } { 5 } , - \frac { 2 \sqrt { 5 } } { 5 } \right)$ is on the unit circle. Find $\sin t$ and $\cos t$

A) $\sin t = - \frac { 2 \sqrt { 5 } } { 5 } , \quad \cos t = \frac { \sqrt { 5 } } { 5 }$
B) $\sin t = - \frac { \sqrt { 5 } } { 5 } , \quad \cos t = \frac { \sqrt { 5 } } { 5 }$
C) $\sin t = \frac { \sqrt { 5 } } { 5 } , \quad \cos t = - \frac { 2 \sqrt { 5 } } { 5 }$
D) $\sin t = 2 \sqrt { 5 } , \cos t = \frac { \sqrt { 5 } } { 5 }$
E) none of these

Question 55

Find to three decimal places the value of

A)
B)
C)
D)
E) none

Question 56

Use the fundamental identities to write the first expression in terms of the second. $\tan ^ { 2 } t \sec t , \cos t$

A) $\frac { 1 - \cos ^ { 2 } t } { \cos ^ { 3 } t }$

B) $\frac { 1 } { \cos t }$

C) $\frac { 1 + \cos ^ { 2 } t } { \sqrt { 1 - \cos t } }$

D) $\frac { \cos ^ { 2 } t - 1 } { \cos ^ { 3 } t }$

E) $\frac { 1 } { \cos ^ { 2 } t }$

Question 57

Find the reference number and the terminal point $p ( x , y )$ determined by $t = - \frac { 11 \pi } { 6 }$

A) $- \frac { \pi } { 6 }$ , $P ( x , y ) = \left( \frac { - \sqrt { 5 } } { 2 } , - \frac { 1 } { 2 } \right)$

B) $\pi$ , $P ( x , y ) = ( - 1,0 )$

C) $\frac { \pi } { 6 }$ , $D ( x , y ) = \left( \frac { \sqrt { 3 } } { 2 } , \frac { 1 } { 2 } \right)$

D) $\frac { \pi } { 3 }$ , $p ( x , y ) = \left( \frac { 1 } { 2 } , \frac { \sqrt { 5 } } { 2 } \right)$

E) $\frac { 11 \pi } { 6 }$ , $p ( x , y ) = \left( - \frac { - \sqrt {5 } } { 2 } , - \frac { 1 } { 2 } \right)$

Question 58

Find the period of the function $y = \frac { 1 } { 5 } \cot x$

A) $\frac { \pi } { 5 }$
B) $5 \pi$
C) $\pi$
D) $\frac { 2 \pi } { 5 }$
E) $\frac { 5 \pi } { 2 }$

Question 59

Suppose that point $P \left( - \frac { 2 \sqrt { 2 } } { 3 } , - \frac { 1 } { 3 } \right)$ is on the unit circle. Find $\sin t$ and $\cos t$

A) $\sin t = - 2 \sqrt { 2 } , \quad \cos t = 3$
B) $\sin t = - 2 \sqrt { 3 } , \cos t = 3$
C) $\sin t = - \frac { 2 \sqrt { 3 } } { 3 } , \cos t = - \frac { 1 } { 3 }$
D) $\sin t = - \frac { 1 } { 3 } , \quad \cos t = - \frac { 2 - \sqrt { 2 } } { 3 }$
E) none of these

Question 60

Graph the function.

A)





B)



C)



E) none
D)

Question 61

Suppose that the terminal point determined by is the point
On the unit circle. Find the terminal point determined by

A)
B)
C)
D)
E)

Question 62

Find the exact value of and

A)
B)
C)
D)
E) none of these

Question 63

Sketch the graph of the function. $y = 2 \cos \left( 4 x - \frac { \pi } { 2 } \right)$

A)
B)
C)
D)
E) none

Question 64

Use a graphing device to find the maximum and minimum values of the function $y = \cos x - \frac { 1 } { 2 } \cos 2 x$

A)maximum value: $\frac { 4 } { 3 }$
, minimum value: $0$

B)maximum value: $\frac { 3 } { 4 }$
, minimum value: $- 1$

C)maximum value: $\frac { 3 } { 4 }$
, minimum value: $- \frac { 3 } { 2 }$

D)maximum value: $\frac { 3 } { 4 }$
, minimum value: $- \frac { 4 } { 3 }$

E) none of these

Question 65

Find the period of the function $y = 3 \sec x$

A) $2 \pi$
B) $\pi$
C) $3 \pi$
D) $\frac { \pi } { 3 }$
E) $\frac { 2 \pi } { 3 }$

Question 66

Find the exact value of $\sec \left( - \frac { 5 \pi } { 2 } \right)$ and $\csc \left( - \frac { 7 \pi } { 2 } \right)$

A) $\sec \left( - \frac { 5 \pi } { 2 } \right) = 0 , \csc \left( - \frac { 7 \pi } { 2 } \right) = 1$

B) $\sec \left( - \frac { 5 \pi } { 2 } \right) = 0 , \csc \left( - \frac { 7 \pi } { 2 } \right) = - 1$

C) $\sec \left( - \frac { 5 \pi } { 2 } \right) = 1 , \csc \left( - \frac { 7 \pi } { 2 } \right) = \text { undefined }$

D) $\sec \left( - \frac { 5 \pi } { 2 } \right) = \text { undefined, } \csc \left( - \frac { 7 \pi } { 2 } \right) = 1$

E) none of these

Question 67

Find the approximate value of using a calculator

A)
B)
C)
D)
E)

Question 68

Use the fundamental identities to write the first expression in terms of the second. $\tan ^ { 2 } t \sec t , \cos t$

A) $\frac { 1 - \cos ^ { 2 } t } { \cos ^ { 3 } t }$

B) $\frac { 1 } { \cos t }$

C) $\frac { 1 + \cos ^ { 2 } t } { \sqrt { 1 - \cos t } }$

D) $\frac { \cos ^ { 2 } t - 1 } { \cos ^ { 3 } t }$

E) $\frac { 1 } { \cos ^ { 3 } t }$

Question 69

The function $y = - 2 \cos \left( \frac { t } { 3 } \right)$ models the displacement of an object moving in simple harmonic motion, where y is measured in inches and t in seconds. Find frequency of motion

A) $\frac { 1 } { 3 \pi }$

B) $\frac { 1 } { 6 \pi }$

C) $6 \pi$
D) $3 \pi$
E) $2 \pi$

Question 70

Find the reference number for $t = - \frac { \pi } { 6 }$

A) $\bar { t } = \frac { 11 \pi } { 6 }$
B) $\bar { t }= - \frac { 5 \pi } { 6 }$
C) $\bar { t }= \frac { \pi } { 6 }$
D) $\bar { t } = \frac { \pi } { 12 }$
E) $\bar { t }= - \frac { \pi } { 6 }$

Question 71

Determine whether the function is even, odd, or neither.

A)even
B)odd
C)neither

Question 72

Find the reference number for $t = - \frac { 22 \pi } { 3 }$

A) $\bar { t } = \frac { \pi } { 6 }$
B) $\bar { t } = - \frac { 7 \pi } { 6 }$
C) $\bar { t } = \frac { \pi } { 3 }$
D) $\bar { t } = - \frac { \pi } { 3 }$
E) $\bar { t } = 7 \pi$

Question 73

If the terminal point determined by $t$ is $\left( \frac { 12 } { 13 } , - \frac { 5 } { 13 } \right)$ , find $\sin t$ , $\cos t$ And $\tan t$

A) $\sin ( t ) = \frac { 12 } { 13 }$ , $\cos ( t ) = \frac { 5 } { 13 }$ , $\tan ( t ) = \frac { 5 } { 12 }$

B) $\sin ( t ) = - \frac { 5 } { 13 }$ , $\cos ( t ) = \frac { 12 } { 13 }$ , $\tan ( t ) = - \frac { 5 } { 12 }$

C) $\sin ( t ) = \frac { 5 } { 13 }$ , $\cos ( t ) = - \frac { 12 } { \Gamma 3 }$ , $\tan ( t ) = - \frac { 13 } { 12 }$

D) $\sin ( t ) = - \frac { 5 } { 13 }$ , $\cos ( t ) = \frac { 12 } { 13 }$ , $\tan ( t ) = - \frac { 12 } { 5 }$

E) $\sin ( t ) = - \frac { 5 } { 13 }$ , $\cos ( t ) = \frac { 12 } { 13 }$ , $\tan t = \frac { 12 } { 5 }$

Question 74

Find the vertical asymptotes for the function in the interval

A)
B)
C)
D)
E)

Question 75

Find the exact value of the expression, if it is defined.

A)
B)
C)
D)
E)

Question 76

The point $P ( x , y )$ is on the unit circle in quadrant IV. If $y = - 5 / 6$ Find x

A) $x = \sqrt { 11 } , 6$
B) $x = - \sqrt { 11 } / 6$
C) $x = 11 / 6$
D) $x = - 11 / 6$
E)none of these

Question 77

Find a function that models simple harmonic motion having the given properties. Assume that the displacement is zero at time .amplitude 10 m, frequency

A)
B)
C)
D)
E)

Question 78

Determine whether the function is even, odd, or neither.

A)even
B)odd
C)neither

Question 79

The point $P ( x , y )$ is on the unit circle in quadrant IV. If $x = \sqrt { 11 } / 6$
find y.

Question 80

Which function has y-axis symmetry?

A)
B)
C)
D)
E) all have y-axis symmetry