Question 1

Find an algebraic expression equivalent to the given expression.
-$\cot \left( \arcsin \frac { \mathrm { u } } { \sqrt { \mathrm { u } ^ { 2 } + 5 } } \right)$

Accepted Answer

A)  $\frac { \sqrt { u ^ { 2 } + 5 } } { 5 }$ 
B)  $\frac { \sqrt { 5 } } { u }$ 
C)  $\frac { \sqrt { 5 u ^ { 2 } + 25 } } { 5 }$ 
D)  $\frac { \mathrm { u } \sqrt { \mathrm { u } ^ { 2 } + 5 } } { \mathrm { u } ^ { 2 } + 5 }$ 
A)  $\frac { \sqrt { u ^ { 2 } + 5 } } { 5 }$ 
B)  $\frac { \sqrt { 5 } } { u }$ 
C)  $\frac { \sqrt { 5 u ^ { 2 } + 25 } } { 5 }$ 
D)  $\frac { \mathrm { u } \sqrt { \mathrm { u } ^ { 2 } + 5 } } { \mathrm { u } ^ { 2 } + 5 }$

Question 2

Evaluate without using a calculator by using ratios in a reference triangle.
-$\tan \frac { - 7 \pi } { 4 }$

Accepted Answer

A)  $\frac { \sqrt { 3 } } { 3 }$ 
B)  $- 1$ 
C)  $\sqrt { 3 }$ 
D)  1 
A)  $\frac { \sqrt { 3 } } { 3 }$ 
B)  $- 1$ 
C)  $\sqrt { 3 }$ 
D)  1

Question 3

Determine the sign (positive or negative) of the given value without use of a calculator.
-$\cos \frac { \pi } { 9 }$

Accepted Answer

A)  Negative 
B)  Positive 
A)  Negative 
B)  Positive

Question 4

Find an algebraic expression equivalent to the given expression.
-$\tan ( \operatorname { arcsec } \mathrm { u } )$

Accepted Answer

A)  $\frac { \mathrm { u } \sqrt { \mathrm { u } ^ { 2 } + 1 } } { \mathrm { u } ^ { 2 } + 1 }$ 
B)  $\frac { \sqrt { u ^ { 2 } - 1 } } { u }$ 
C)  $\sqrt { u ^ { 2 } + 1 }$ 
D)  $\sqrt { u ^ { 2 } - 1 }$ 
A)  $\frac { \mathrm { u } \sqrt { \mathrm { u } ^ { 2 } + 1 } } { \mathrm { u } ^ { 2 } + 1 }$ 
B)  $\frac { \sqrt { u ^ { 2 } - 1 } } { u }$ 
C)  $\sqrt { u ^ { 2 } + 1 }$ 
D)  $\sqrt { u ^ { 2 } - 1 }$

Question 5

Determine whether the function is periodic. If it is periodic, find the period.
-$$f ( x ) = 7 \sin ^ { 3 } x$$

Accepted Answer

A)  Not periodic 
B)  $\pi$ 
C)  $2 \pi$ 
D)  $4 \pi$ 
A)  Not periodic 
B)  $\pi$ 
C)  $2 \pi$ 
D)  $4 \pi$

Question 6

Solve the problem.
-Find the distance in kilometers between the cities whose latitudes are given. Assume that the cities are on a north-south line and that the radius of the earth is $6400 \mathrm {~km}$. City E, $47 ^ { \circ } \mathrm { N }$ and city F, $80 ^ { \circ } \mathrm { S }$

Accepted Answer

A)  $3678 \mathrm {~km}$ 
B)  $14,194 \mathrm {~km}$ 
C)  $3686 \mathrm {~km}$ 
D)  $14,186 \mathrm {~km}$ 
A)  $3678 \mathrm {~km}$ 
B)  $14,194 \mathrm {~km}$ 
C)  $3686 \mathrm {~km}$ 
D)  $14,186 \mathrm {~km}$

Question 7

Determine whether the given function is positive or negative for values of t in the specified quadrant.
-Quadrant IV, $\sin t$

Accepted Answer

A)  Negative 
B)  Positive 
A)  Negative 
B)  Positive

Question 8

Describe the transformation required to obtain the graph of the given function from the basic inverse trigonometric
graph.
-$f ( x ) = \arcsin \left( \frac { x } { 5 } \right)$

Accepted Answer

A)  Vertical translation up 5 units 
B)  Horizontal shrink by a factor of $\frac { 1 } { 5 }$ 
C)  Vertical stretch by a factor of 5 
D)  Horizontal stretch by a factor of 5 
A)  Vertical translation up 5 units 
B)  Horizontal shrink by a factor of $\frac { 1 } { 5 }$ 
C)  Vertical stretch by a factor of 5 
D)  Horizontal stretch by a factor of 5

Question 9

Use a calculator to find the approximate value of the expression. Express your answer in radians and round to three
decimal places.
-$\sin ^ { - 1 } ( - 0.4653 )$

Accepted Answer

A)  $2.658$ 
B)  $3.626$ 
C)  $- 0.484$ 
D)  $2.055$ 
A)  $2.658$ 
B)  $3.626$ 
C)  $- 0.484$ 
D)  $2.055$

Question 10

Find the exact value of the composition.
-$\arccos [ \sin ( \pi / 6 ) ]$

Accepted Answer

A)  $\frac { \pi } { 3 }$ 
B)  $\frac { 1 } { 2 }$ 
C)  $\frac { \pi } { 6 }$ 
D)  $\frac { \sqrt { 3 } } { 2 }$ 
A)  $\frac { \pi } { 3 }$ 
B)  $\frac { 1 } { 2 }$ 
C)  $\frac { \pi } { 6 }$ 
D)  $\frac { \sqrt { 3 } } { 2 }$

Question 11

Solve the problem.
-A wheel with a 32 -inch radius is marked at two points on the rim. The distance between the marks along the wheel is found to be 7 inches. What is the angle (to the nearest tenth of a degree) between the radii to the two marks?

Accepted Answer

A)  $12.5 ^ { \circ }$ 
B)  $8.5 ^ { \circ }$ 
C)  $14.5 ^ { \circ }$ 
D)  $10.5 ^ { \circ }$ 
A)  $12.5 ^ { \circ }$ 
B)  $8.5 ^ { \circ }$ 
C)  $14.5 ^ { \circ }$ 
D)  $10.5 ^ { \circ }$

Question 12

Convert the angle to decimal degrees and round to the nearest hundredth of a degree.
-$212 ^ { \circ } 21 ^ { \prime } 53 ^ { \prime \prime }$

Accepted Answer

A)  $212.37 ^ { \circ }$ 
B)  $212.32 ^ { \circ }$ 
C)  $212.42 ^ { \circ }$ 
D)  $212.36 ^ { \circ }$ 
A)  $212.37 ^ { \circ }$ 
B)  $212.32 ^ { \circ }$ 
C)  $212.42 ^ { \circ }$ 
D)  $212.36 ^ { \circ }$

Question 13

Point P is on the terminal side of θ . Evaluate the six trigonometric functions for θ. If the function is undefined, write "undefined."
-$\mathrm { P } ( 12,16 )$; find $\cos \theta$

Accepted Answer

A)  $\frac { 3 } { 5 }$ 
B)  $\frac { 3 } { 4 }$ 
C)  $\frac { 4 } { 5 }$ 
D)  $\frac { 4 } { 3 }$ 
A)  $\frac { 3 } { 5 }$ 
B)  $\frac { 3 } { 4 }$ 
C)  $\frac { 4 } { 5 }$ 
D)  $\frac { 4 } { 3 }$

Question 14

Identify the maximum and minimum values of the function in the interval $ [-2 \pi, 2 \pi] $. Use your understanding of transformations, not your graphing calculator.
-$$y = - \cos 3 x$$

Accepted Answer

A)  Maximum value $= \frac { 1 } { 2 } ;$ minimum value $= - \frac { 1 } { 2 }$ 
B)  Maximum value $= 1$; minimum value $= 0$ 
C)  Maximum value $= 3 ;$ minimum value $= - 3$ 
D)  Maximum value $= 1$; minimum value $= - 1$ 
A)  Maximum value $= \frac { 1 } { 2 } ;$ minimum value $= - \frac { 1 } { 2 }$ 
B)  Maximum value $= 1$; minimum value $= 0$ 
C)  Maximum value $= 3 ;$ minimum value $= - 3$ 
D)  Maximum value $= 1$; minimum value $= - 1$

Question 15

Solve the problem.
-The weekly sales in thousands of items of a product has a seasonal sales record approximated by $\mathrm { n } = 73.92 + 27.7 \sin \frac { \pi \mathrm { t } } { 24 }$ ( $\mathrm { t } =$ time in weeks with $\mathrm { t } = 1$ referring to the first week in the year). During which week(s) will the sales equal 87,770 items?

Accepted Answer

A)  week 4 and week 47 
B)  week 30 and week 47 
C)  week 21 and week 30 
D)  week 4 , week 20 , and week 52 
A)  week 4 and week 47 
B)  week 30 and week 47 
C)  week 21 and week 30 
D)  week 4 , week 20 , and week 52

Question 16

Determine whether the function is periodic. If it is periodic, find the period.
-$$f ( x ) = 8 \sin 2 x + 9 \cos 2 x$$

Accepted Answer

A)  $2 \pi$ 
B)  $\frac { \pi } { 2 }$ 
C)  Not periodic 
D)  $\pi$ 
A)  $2 \pi$ 
B)  $\frac { \pi } { 2 }$ 
C)  Not periodic 
D)  $\pi$

Question 17

Solve the problem.
-The position of a weight attached to a spring is $s ( t ) = - 5 \cos 7 t$. What are the frequency and period of the system?

Accepted Answer

A)  Frequency $= 7$ cycles per sec, period $= \frac { 1 } { 7 } \mathrm { sec }$ 
B)  Frequency $= \frac { \pi } { 7 }$ cycles per sec, period $= \frac { 7 } { \pi } \mathrm { sec }$ 
C)  Frequency $= \frac { 7 } { \pi }$ cycles per sec, period $= \frac { \pi } { 7 } \mathrm { sec }$ 
D)  Frequency $= \frac { 7 } { 2 \pi }$ cycles per sec, period $= \frac { 2 \pi } { 7 } \mathrm { sec }$ 
A)  Frequency $= 7$ cycles per sec, period $= \frac { 1 } { 7 } \mathrm { sec }$ 
B)  Frequency $= \frac { \pi } { 7 }$ cycles per sec, period $= \frac { 7 } { \pi } \mathrm { sec }$ 
C)  Frequency $= \frac { 7 } { \pi }$ cycles per sec, period $= \frac { \pi } { 7 } \mathrm { sec }$ 
D)  Frequency $= \frac { 7 } { 2 \pi }$ cycles per sec, period $= \frac { 2 \pi } { 7 } \mathrm { sec }$

Question 18

Choose the graph which matches the function.
-$f ( x ) = \left( \frac { 1 } { 2 } \right) ^ { x } \sin 3 x$

Accepted Answer

A)  Damping factor $= 3$, damping occurs as $x \rightarrow 0$ 
B)  Damping factor $= \left( \frac { 1 } { 2 } \right) ^ { x }$, damping occurs as $x \rightarrow \infty$ 
C)  No damping 
D)  Damping factor $= \left( \frac { 1 } { 2 } \right) ^ { x }$, damping occurs as $x \rightarrow 0$ 
A)  Damping factor $= 3$, damping occurs as $x \rightarrow 0$ 
B)  Damping factor $= \left( \frac { 1 } { 2 } \right) ^ { x }$, damping occurs as $x \rightarrow \infty$ 
C)  No damping 
D)  Damping factor $= \left( \frac { 1 } { 2 } \right) ^ { x }$, damping occurs as $x \rightarrow 0$

Question 19

Solve the problem.
-A pendulum of length $14.8$ inches swings $2 ^ { \circ } 18 ^ { \prime }$ to each side of its vertical position. To the nearest hundredth of an inch, what is the length of the arc through which the end of the pendulum swings?

Accepted Answer

A)  $0.89 \mathrm { in }$. 
B)  $1.09 \mathrm { in } .$ 
C)  $1.19 \mathrm { in }$. 
D)  $0.99 \mathrm { in }$. 
A)  $0.89 \mathrm { in }$. 
B)  $1.09 \mathrm { in } .$ 
C)  $1.19 \mathrm { in }$. 
D)  $0.99 \mathrm { in }$.

Question 20

Use the arc length formula and the given information to find the indicated quantity.
-$\mathrm { s } = 15 \mathrm {~cm} , \theta = 45 ^ { \circ }$; find $\mathrm { r }$

Accepted Answer

A)  $\frac { 1 } { 3 } \mathrm {~cm}$ 
B)  $\frac { 30 } { \pi } \mathrm { cm }$ 
C)  $\frac { 120 } { \pi } \mathrm { cm }$ 
D)  $\frac { 60 } { \pi } \mathrm { cm }$ 
A)  $\frac { 1 } { 3 } \mathrm {~cm}$ 
B)  $\frac { 30 } { \pi } \mathrm { cm }$ 
C)  $\frac { 120 } { \pi } \mathrm { cm }$ 
D)  $\frac { 60 } { \pi } \mathrm { cm }$

Find an algebraic expression equivalent to the given expression. - $\cot \left( \arcsin \frac { \mathrm { u } } { \sqrt { \mathrm { u } ^ { 2 } + 5 } } \right)$

Evaluate without using a calculator by using ratios in a reference triangle. - $\tan \frac { - 7 \pi } { 4 }$

Determine the sign (positive or negative) of the given value without use of a calculator. - $\cos \frac { \pi } { 9 }$

Find an algebraic expression equivalent to the given expression. - $\tan ( \operatorname { arcsec } \mathrm { u } )$

Determine whether the function is periodic. If it is periodic, find the period. - $f ( x ) = 7 \sin ^ { 3 } x$

Solve the problem. -Find the distance in kilometers between the cities whose latitudes are given. Assume that the cities are on a north-south line and that the radius of the earth is $6400 \mathrm {~km}$ . City E, $47 ^ { \circ } \mathrm { N }$ and city F, $80 ^ { \circ } \mathrm { S }$

Determine whether the given function is positive or negative for values of t in the specified quadrant. -Quadrant IV, $\sin t$

Describe the transformation required to obtain the graph of the given function from the basic inverse trigonometric graph. - $f ( x ) = \arcsin \left( \frac { x } { 5 } \right)$

Use a calculator to find the approximate value of the expression. Express your answer in radians and round to three decimal places. - $\sin ^ { - 1 } ( - 0.4653 )$

Find the exact value of the composition. - $\arccos [ \sin ( \pi / 6 ) ]$

Solve the problem. -A wheel with a 32 -inch radius is marked at two points on the rim. The distance between the marks along the wheel is found to be 7 inches. What is the angle (to the nearest tenth of a degree) between the radii to the two marks?

Convert the angle to decimal degrees and round to the nearest hundredth of a degree. - $212 ^ { \circ } 21 ^ { \prime } 53 ^ { \prime \prime }$

Point P is on the terminal side of θ . Evaluate the six trigonometric functions for θ. If the function is undefined, write "undefined." - $\mathrm { P } ( 12,16 )$ ; find $\cos \theta$

Identify the maximum and minimum values of the function in the interval $[-2 \pi, 2 \pi]$ . Use your understanding of transformations, not your graphing calculator. - $y = - \cos 3 x$

Determine whether the function is periodic. If it is periodic, find the period. - $f ( x ) = 8 \sin 2 x + 9 \cos 2 x$

Solve the problem. -The position of a weight attached to a spring is $s ( t ) = - 5 \cos 7 t$ . What are the frequency and period of the system?

Choose the graph which matches the function. - $f ( x ) = \left( \frac { 1 } { 2 } \right) ^ { x } \sin 3 x$

Solve the problem. -A pendulum of length $14.8$ inches swings $2 ^ { \circ } 18 ^ { \prime }$ to each side of its vertical position. To the nearest hundredth of an inch, what is the length of the arc through which the end of the pendulum swings?

Use the arc length formula and the given information to find the indicated quantity. - $\mathrm { s } = 15 \mathrm {~cm} , \theta = 45 ^ { \circ }$ ; find $\mathrm { r }$

Functions and Graphs

Polynomial, Power, and Rational Functions

Exponential, Logistic, and Logarithmic Functions

Analytic Trigonometry

Applications of Trigonometry

Systems and Matrices

Analytic Geometry in Two and Three Dimensions

Discrete Mathematics

Statistics and Probability

An Introduction to Calculus: Limits, Derivatives, and Integrals

Prerequisites

Filters

Exam 4: Trigonometric Functions

Find an algebraic expression equivalent to the given expression. - cot⁡(arcsin⁡uu2+5)\cot \left( \arcsin \frac { \mathrm { u } } { \sqrt { \mathrm { u } ^ { 2 } + 5 } } \right)cot(arcsinu2+5​u​)

Evaluate without using a calculator by using ratios in a reference triangle. - tan⁡−7π4\tan \frac { - 7 \pi } { 4 }tan4−7π​

Determine the sign (positive or negative) of the given value without use of a calculator. - cos⁡π9\cos \frac { \pi } { 9 }cos9π​

Find an algebraic expression equivalent to the given expression. - tan⁡(arcsec⁡u)\tan ( \operatorname { arcsec } \mathrm { u } )tan(arcsecu)

Determine whether the function is periodic. If it is periodic, find the period. - f(x)=7sin⁡3xf ( x ) = 7 \sin ^ { 3 } xf(x)=7sin3x

Determine whether the given function is positive or negative for values of t in the specified quadrant. -Quadrant IV, sin⁡t\sin tsint

Describe the transformation required to obtain the graph of the given function from the basic inverse trigonometric graph. - f(x)=arcsin⁡(x5)f ( x ) = \arcsin \left( \frac { x } { 5 } \right)f(x)=arcsin(5x​)

Use a calculator to find the approximate value of the expression. Express your answer in radians and round to three decimal places. - sin⁡−1(−0.4653)\sin ^ { - 1 } ( - 0.4653 )sin−1(−0.4653)

Find the exact value of the composition. - arccos⁡[sin⁡(π/6)]\arccos [ \sin ( \pi / 6 ) ]arccos[sin(π/6)]