Question 1

Determine whether the graph of f(x) is a sinusoid.
-$$f ( x ) = \sin 17 x + \cos 2 x$$

Accepted Answer

A) True 
 B)False

Question 2

Solve the problem.
-Use a tangent function to estimate the grade resistance of a $2500 - \mathrm { lb }$ car travelling on a $- 3.5 ^ { \circ }$ grade. Round your answer to the nearest pound.

Accepted Answer

A)  $- 153$ pounds 
B)  936 pounds 
C)  468 pounds 
D)  306 pounds 
A)  $- 153$ pounds 
B)  936 pounds 
C)  468 pounds 
D)  306 pounds

Question 3

Evaluate without using a calculator by using ratios in a reference triangle.
-$\csc 240 ^ { \circ }$

Accepted Answer

A)  $- \frac { 2 \sqrt { 3 } } { 3 }$ 
B)  $- 2$ 
C)  $\frac { 2 \sqrt { 3 } } { 3 }$ 
D)  2 
A)  $- \frac { 2 \sqrt { 3 } } { 3 }$ 
B)  $- 2$ 
C)  $\frac { 2 \sqrt { 3 } } { 3 }$ 
D)  2

Question 4

Solve the problem.
-A stretch of highway has a 2-ft vertical rise for every $100 \mathrm { ft }$ of horizontal run. Use a function involving tangent to estimate the grade resistance for a 2200-lb car travelling on this stretch of highway. Give your answer to the nearest pound.

Accepted Answer

A)  20 pounds 
B)  44 pounds 
C)  11 pounds 
D)  24 pounds 
A)  20 pounds 
B)  44 pounds 
C)  11 pounds 
D)  24 pounds

Question 5

Evaluate the trigonometric function of the given quadrantal angle. If the value is undefined, write "undefined."
-$\sin ( - 11 \pi )$

Accepted Answer

A)  0 
B)  undefined 
C)  1 
D)  $- 1$ 
A)  0 
B)  undefined 
C)  1 
D)  $- 1$

Question 6

Solve the problem.
-A guitar string is plucked so that it vibrates with a frequency of $\mathrm { F } = 67$. Suppose the maximum displacement at the center of the string is $s ( 0 ) = 0.51$. Find an equation of the form $s ( t ) = a$ cos bt to model this displacement. Round constants to 2 decimal places.

Accepted Answer

A)  $s ( t ) = 0.51 \cos 420.97 \mathrm { t }$ 
B)  $s ( t ) = 0.51 \cos 10.66 t$ 
C)  $s ( t ) = 1.02 \cos 420.97 \mathrm { t }$ 
D)  $s ( t ) = 1.02 \cos 10.66 \mathrm { t }$ 
A)  $s ( t ) = 0.51 \cos 420.97 \mathrm { t }$ 
B)  $s ( t ) = 0.51 \cos 10.66 t$ 
C)  $s ( t ) = 1.02 \cos 420.97 \mathrm { t }$ 
D)  $s ( t ) = 1.02 \cos 10.66 \mathrm { t }$

Question 7

Find an algebraic expression equivalent to the given expression.
-$\sec \left( \operatorname { arccot } \frac { \sqrt { 5 } } { u } \right)$

Accepted Answer

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

Question 8

Solve the problem.
-A person is watching a boat from the top of a lighthouse. The boat is approaching the lighthouse directly. When first noticed the angle of depression to the boat is $15 ^ { \circ } 57 ^ { \prime }$. When the boat stops, the angle of depression is $55 ^ { \circ } 5 ^ { \prime }$ . The lighthouse is 200 feet tall. How far did the boat travel from when it was first noticed until it stopped? Round your answer to the hundredths place.

Accepted Answer

A)  $560.18 \mathrm { ft }$ 
B)  $605.38 \mathrm { ft }$ 
C)  $544.88 \mathrm { ft }$ 
D)  $582.68 \mathrm { ft }$ 
A)  $560.18 \mathrm { ft }$ 
B)  $605.38 \mathrm { ft }$ 
C)  $544.88 \mathrm { ft }$ 
D)  $582.68 \mathrm { ft }$

Question 9

Find a, b, and h so that f(x) a sin (b(x - h)). Round to two decimal places if necessary.
-$$f ( x ) = 2 \sin \pi x + 3 \cos \pi x$$

Accepted Answer

A)  $a = 0.31 , b = 3.14 , h = 3.61$ 
B)  $\mathrm { a } = 3.61 , \mathrm {~b} = 3.14 , \mathrm {~h} = - 0.31$ 
C)  $a = 5.00 , b = 6.28 , h = - 0.45$ 
D)  $\mathrm { a } = 4.29 , \mathrm {~b} = 6.28 , \mathrm {~h} = 0.31$ 
A)  $a = 0.31 , b = 3.14 , h = 3.61$ 
B)  $\mathrm { a } = 3.61 , \mathrm {~b} = 3.14 , \mathrm {~h} = - 0.31$ 
C)  $a = 5.00 , b = 6.28 , h = - 0.45$ 
D)  $\mathrm { a } = 4.29 , \mathrm {~b} = 6.28 , \mathrm {~h} = 0.31$

Question 10

Find the specified quantity.
-Find the amplitude of $y = - 4 \sin ( 3 x + \pi )$.

Accepted Answer

A)  $\pi$ 
B)  3 
C)  $- 12$ 
D)  4 
A)  $\pi$ 
B)  3 
C)  $- 12$ 
D)  4

Question 11

Describe the transformation required to obtain the graph of the given function from the basic trigonometric graph.
-$$y = 3 \cot \pi x$$

Accepted Answer

A)  Vertical stretch by a factor of $\frac { 1 } { \pi }$ and a horizontal shrink by a factor of 3 
B)  Vertical translation up 3 units and horizontal translation to the right $\pi$ units 
C)  Vertical translation down 3 units and horizontal translation to the left $\pi$ units 
D)  Vertical stretch by a factor of 3 and a horizontal shrink by a factor of $\frac { 1 } { \pi }$ 
A)  Vertical stretch by a factor of $\frac { 1 } { \pi }$ and a horizontal shrink by a factor of 3 
B)  Vertical translation up 3 units and horizontal translation to the right $\pi$ units 
C)  Vertical translation down 3 units and horizontal translation to the left $\pi$ units 
D)  Vertical stretch by a factor of 3 and a horizontal shrink by a factor of $\frac { 1 } { \pi }$

Question 12

Convert the radian measure to degree measure. Use the value of found on a calculator and round answers to two
decimal places.
-$3 \pi$

Accepted Answer

A)  $270 ^ { \circ }$ 
B)  $540 ^ { \circ }$ 
C)  $1080 \pi ^ { \circ }$ 
D)  $9.42 ^ { \circ }$ 
A)  $270 ^ { \circ }$ 
B)  $540 ^ { \circ }$ 
C)  $1080 \pi ^ { \circ }$ 
D)  $9.42 ^ { \circ }$

Question 13

Solve the problem.
-The path of a projectile fired at an inclination $\theta$ to the horizontal with an initial speed $v _ { 0 }$ is a parabola. The range $R$ of the projectile, the horizontal distance that the projectile travels, is found by the formula
$\mathrm { R } = \frac { \mathrm { v } _ { \mathrm { O } } ^ { 2 } \sin 2 \theta } { \mathrm { g } }$ where $\mathrm { g } = 32.2$ feet per second per second or $\mathrm { g } = 9.8$ meters per second per second. Find the range of a projectile fired with an initial velocity of 127 feet per second at an angle of $27 ^ { \circ }$ to the horizontal. Round your answer to two decimal places.

Accepted Answer

A)  $454.81 \mathrm { ft }$ 
B)  $405.24 \mathrm { ft }$ 
C)  $227.40 \mathrm { ft }$ 
D)  $405.14 \mathrm { ft }$ 
A)  $454.81 \mathrm { ft }$ 
B)  $405.24 \mathrm { ft }$ 
C)  $227.40 \mathrm { ft }$ 
D)  $405.14 \mathrm { ft }$

Question 14

Solve the problem.
-The position of a weight attached to a spring is $s ( t ) = - 3 \cos 6 \pi$. What is the maximum height that the weight rises above the equilibrium position?

Accepted Answer

A)  3 
B)  6 
C)  $\frac { 6 } { \pi }$ 
D)  $0.5$ 
A)  3 
B)  6 
C)  $\frac { 6 } { \pi }$ 
D)  $0.5$

Question 15

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.
-$$y = \frac { 1 } { 2 } \sin 2 x$$

Accepted Answer

A)  Maximum value $= \frac { 1 } { 2 } ;$ minimum value $= - \frac { 1 } { 2 }$ 
B)  Maximum value $= \frac { 1 } { 2 } ;$ minimum value $= 0$ 
C)  Maximum value $= \frac { 1 } { 4 } ;$ minimum value $= - \frac { 1 } { 4 }$ 
D)  Maximum value $= 2$; minimum value $= - 2$ 
A)  Maximum value $= \frac { 1 } { 2 } ;$ minimum value $= - \frac { 1 } { 2 }$ 
B)  Maximum value $= \frac { 1 } { 2 } ;$ minimum value $= 0$ 
C)  Maximum value $= \frac { 1 } { 4 } ;$ minimum value $= - \frac { 1 } { 4 }$ 
D)  Maximum value $= 2$; minimum value $= - 2$

Question 16

Use a calculator to solve for x in the given interval
-$\cot \theta = 3.064,0 ^ { \circ } \leq \theta \leq 90 ^ { \circ }$

Accepted Answer

A)  $18.08 ^ { \circ }$ 
B)  $70.95 ^ { \circ }$ 
C)  $- 18.08 ^ { \circ }$ 
D)  $19.05 ^ { \circ }$ 
A)  $18.08 ^ { \circ }$ 
B)  $70.95 ^ { \circ }$ 
C)  $- 18.08 ^ { \circ }$ 
D)  $19.05 ^ { \circ }$

Question 17

Find the value of the unique real number between 0 and 2 that satisfies the given conditions.
-$\tan \theta = - \frac { \sqrt { 3 } } { 3 }$ and $\cos \theta < 0$

Accepted Answer

A)  $\frac { \pi } { 6 }$ 
B)  $\frac { 5 \pi } { 6 }$ 
C)  $\frac { 2 \pi } { 3 }$ 
D)  $\frac { 7 \pi } { 6 }$ 
A)  $\frac { \pi } { 6 }$ 
B)  $\frac { 5 \pi } { 6 }$ 
C)  $\frac { 2 \pi } { 3 }$ 
D)  $\frac { 7 \pi } { 6 }$

Question 18

Use a calculator to graph the function.
-$$y = | \sin 2 x |$$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 19

Describe the transformation required to obtain the graph of the given function from the basic trigonometric graph.
-y = csc x - 2

Accepted Answer

A)  Vertical translation down 2 units 
B)  Horizontal translation to the right 2 units 
C)  Vertical stretch by a factor of 2 
D)  Vertical translation up 2 units 
A)  Vertical translation down 2 units 
B)  Horizontal translation to the right 2 units 
C)  Vertical stretch by a factor of 2 
D)  Vertical translation up 2 units

Question 20

Solve the problem.
-The position of a weight attached to a spring is $\mathrm { s } ( \mathrm { t } ) = - 4 \cos 7 \pi \mathrm { t }$ inches after $t$ seconds. When does the weight first reach its maximum height?

Accepted Answer

A)  After $\frac { 7 } { \pi } \mathrm { sec }$ 
B)  After $\frac { \pi } { 7 } \mathrm { sec }$ 
C)  After $\frac { 7 } { 4 } \mathrm { sec }$ 
D)  After $\frac { 1 } { 7 } \mathrm { sec }$ 
A)  After $\frac { 7 } { \pi } \mathrm { sec }$ 
B)  After $\frac { \pi } { 7 } \mathrm { sec }$ 
C)  After $\frac { 7 } { 4 } \mathrm { sec }$ 
D)  After $\frac { 1 } { 7 } \mathrm { sec }$

Determine whether the graph of f(x) is a sinusoid. - $f ( x ) = \sin 17 x + \cos 2 x$

Solve the problem. -Use a tangent function to estimate the grade resistance of a $2500 - \mathrm { lb }$ car travelling on a $- 3.5 ^ { \circ }$ grade. Round your answer to the nearest pound.

Evaluate without using a calculator by using ratios in a reference triangle. - $\csc 240 ^ { \circ }$

Solve the problem. -A stretch of highway has a 2-ft vertical rise for every $100 \mathrm { ft }$ of horizontal run. Use a function involving tangent to estimate the grade resistance for a 2200-lb car travelling on this stretch of highway. Give your answer to the nearest pound.

Evaluate the trigonometric function of the given quadrantal angle. If the value is undefined, write "undefined." - $\sin ( - 11 \pi )$

Find an algebraic expression equivalent to the given expression. - $\sec \left( \operatorname { arccot } \frac { \sqrt { 5 } } { u } \right)$

Find a, b, and h so that f(x) a sin (b(x - h)). Round to two decimal places if necessary. - $f ( x ) = 2 \sin \pi x + 3 \cos \pi x$

Find the specified quantity. -Find the amplitude of $y = - 4 \sin ( 3 x + \pi )$ .

Describe the transformation required to obtain the graph of the given function from the basic trigonometric graph. - $y = 3 \cot \pi x$

Convert the radian measure to degree measure. Use the value of found on a calculator and round answers to two decimal places. - $3 \pi$

Solve the problem. -The position of a weight attached to a spring is $s ( t ) = - 3 \cos 6 \pi$ . What is the maximum height that the weight rises above the equilibrium position?

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. - $y = \frac { 1 } { 2 } \sin 2 x$

Use a calculator to solve for x in the given interval - $\cot \theta = 3.064,0 ^ { \circ } \leq \theta \leq 90 ^ { \circ }$

Find the value of the unique real number between 0 and 2 that satisfies the given conditions. - $\tan \theta = - \frac { \sqrt { 3 } } { 3 }$ and $\cos \theta < 0$

Use a calculator to graph the function. - $y = | \sin 2 x |$ $Use a calculator to graph the function. - y = | \sin 2 x |$

Describe the transformation required to obtain the graph of the given function from the basic trigonometric graph. -y = csc x - 2

Solve the problem. -The position of a weight attached to a spring is $\mathrm { s } ( \mathrm { t } ) = - 4 \cos 7 \pi \mathrm { t }$ inches after $t$ seconds. When does the weight first reach its maximum height?

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

Determine whether the graph of f(x) is a sinusoid. - f(x)=sin⁡17x+cos⁡2xf ( x ) = \sin 17 x + \cos 2 xf(x)=sin17x+cos2x

Solve the problem. -Use a tangent function to estimate the grade resistance of a 2500−lb2500 - \mathrm { lb }2500−lb car travelling on a −3.5∘- 3.5 ^ { \circ }−3.5∘ grade. Round your answer to the nearest pound.

Evaluate without using a calculator by using ratios in a reference triangle. - csc⁡240∘\csc 240 ^ { \circ }csc240∘

Solve the problem. -A stretch of highway has a 2-ft vertical rise for every 100ft100 \mathrm { ft }100ft of horizontal run. Use a function involving tangent to estimate the grade resistance for a 2200-lb car travelling on this stretch of highway. Give your answer to the nearest pound.

Evaluate the trigonometric function of the given quadrantal angle. If the value is undefined, write "undefined." - sin⁡(−11π)\sin ( - 11 \pi )sin(−11π)

Find an algebraic expression equivalent to the given expression. - sec⁡(arccot⁡5u)\sec \left( \operatorname { arccot } \frac { \sqrt { 5 } } { u } \right)sec(arccotu5​​)

Find a, b, and h so that f(x) a sin (b(x - h)). Round to two decimal places if necessary. - f(x)=2sin⁡πx+3cos⁡πxf ( x ) = 2 \sin \pi x + 3 \cos \pi xf(x)=2sinπx+3cosπx

Find the specified quantity. -Find the amplitude of y=−4sin⁡(3x+π)y = - 4 \sin ( 3 x + \pi )y=−4sin(3x+π) .

Describe the transformation required to obtain the graph of the given function from the basic trigonometric graph. - y=3cot⁡πxy = 3 \cot \pi xy=3cotπx

Convert the radian measure to degree measure. Use the value of found on a calculator and round answers to two decimal places. - 3π3 \pi3π

Solve the problem. -The position of a weight attached to a spring is s(t)=−3cos⁡6πs ( t ) = - 3 \cos 6 \pis(t)=−3cos6π . What is the maximum height that the weight rises above the equilibrium position?

Identify the maximum and minimum values of the function in the interval [−2π,2π] [-2 \pi, 2 \pi] [−2π,2π] . Use your understanding of transformations, not your graphing calculator. - y=12sin⁡2xy = \frac { 1 } { 2 } \sin 2 xy=21​sin2x

Use a calculator to solve for x in the given interval - cot⁡θ=3.064,0∘≤θ≤90∘\cot \theta = 3.064,0 ^ { \circ } \leq \theta \leq 90 ^ { \circ }cotθ=3.064,0∘≤θ≤90∘

Find the value of the unique real number between 0 and 2 that satisfies the given conditions. - tan⁡θ=−33\tan \theta = - \frac { \sqrt { 3 } } { 3 }tanθ=−33​​ and cos⁡θ<0\cos \theta < 0cosθ<0

Use a calculator to graph the function. - y=∣sin⁡2x∣y = | \sin 2 x |y=∣sin2x∣