Question 1

Solve the problem.
-A force of $35 \mathrm { lb }$ is required to hold a $78 \mathrm { lb }$ toolbox on an incline. What angle does the incline make with the horizontal?

Accepted Answer

A)  $77.0 ^ { \circ }$ 
B)  $26.7 ^ { \circ }$ 
C)  $63.3 ^ { \circ }$ 
D)  $13.0 ^ { \circ }$ 
A)  $77.0 ^ { \circ }$ 
B)  $26.7 ^ { \circ }$ 
C)  $63.3 ^ { \circ }$ 
D)  $13.0 ^ { \circ }$

Question 2

Find the polar coordinates of the point(s) of intersection of the given curves for 0 ≤ θ < 2π.
-$$\mathrm { r } = 4 \cos \theta , \mathrm { r } = - 1 + 2 \cos \theta$$

Accepted Answer

A)  $\left( 2 , \frac { 5 \pi } { 6 } \right) , \left( 2 , \frac { \pi } { 6 } \right)$ 
B)  $\left( 2 , \frac { 2 \pi } { 3 } \right) , \left( 2 , \frac { 4 \pi } { 3 } \right)$ 
C)  $\left( - 2 , \frac { 5 \pi } { 3 } \right) , \left( - 2 , \frac { \pi } { 3 } \right)$ 
D)  $\left\{ 2 , \frac { 2 \pi } { 3 } \right) , \left( - 2 , \frac { 4 \pi } { 3 } \right)$ 
A)  $\left( 2 , \frac { 5 \pi } { 6 } \right) , \left( 2 , \frac { \pi } { 6 } \right)$ 
B)  $\left( 2 , \frac { 2 \pi } { 3 } \right) , \left( 2 , \frac { 4 \pi } { 3 } \right)$ 
C)  $\left( - 2 , \frac { 5 \pi } { 3 } \right) , \left( - 2 , \frac { \pi } { 3 } \right)$ 
D)  $\left\{ 2 , \frac { 2 \pi } { 3 } \right) , \left( - 2 , \frac { 4 \pi } { 3 } \right)$

Question 3

Determine whether the pair of vectors is orthogonal.
-$\langle 3,1 \rangle , \langle - 6,36 \rangle$

Accepted Answer

A)  Yes 
B)  No 
A)  Yes 
B)  No

Question 4

Find the quotient and write in rectangular form. First convert the numerator and denominator to trigonometric form.
-$$\frac { 5 \left( \cos 200 ^ { \circ } + i \sin 200 ^ { \circ } \right) } { 4 \left( \cos 50 ^ { \circ } + i \sin 50 ^ { \circ } \right) }$$

Accepted Answer

A)  $- \frac { 1 } { 2 } + \frac { \sqrt { 3 } } { 2 } \mathrm { i }$ 
B)  $- 10 + 10 \sqrt { 3 } \mathrm { i }$ 
C)  $- \frac { 5 \sqrt { 3 } } { 8 } + \frac { 5 } { 8 } \mathrm { i }$ 
D)  $- 2 + 2 \sqrt { 3 } \mathrm { i }$ 
A)  $- \frac { 1 } { 2 } + \frac { \sqrt { 3 } } { 2 } \mathrm { i }$ 
B)  $- 10 + 10 \sqrt { 3 } \mathrm { i }$ 
C)  $- \frac { 5 \sqrt { 3 } } { 8 } + \frac { 5 } { 8 } \mathrm { i }$ 
D)  $- 2 + 2 \sqrt { 3 } \mathrm { i }$

Question 5

Find the given power. Write answer in rectangular form.
-$$[ - 3 + 3 i \sqrt { 3 } ] ^ { 3 }$$

Accepted Answer

A)  $27 + 9 i \sqrt { 3 }$ 
B)  216 
C)  $- 3 + 3 i \sqrt { 3 }$ 
D)  27 
A)  $27 + 9 i \sqrt { 3 }$ 
B)  216 
C)  $- 3 + 3 i \sqrt { 3 }$ 
D)  27

Question 6

Solve the problem.
-Two forces of 575 newtons and 271 newtons act at a point. The resultant force is 657 newtons. Find the angle between the forces.

Accepted Answer

A)  $95.1 ^ { \circ }$ 
B)  $84.9 ^ { \circ }$ 
C)  $101.3 ^ { \circ }$ 
D)  $163.5 ^ { \circ }$ 
A)  $95.1 ^ { \circ }$ 
B)  $84.9 ^ { \circ }$ 
C)  $101.3 ^ { \circ }$ 
D)  $163.5 ^ { \circ }$

Question 7

Write the complex number in rectangular form.
-$6 \left( \cos 330 ^ { \circ } + i \sin 330 ^ { \circ } \right)$

Accepted Answer

A)  $- 3 \sqrt { 3 } - 3 i$ 
B)  $- 3 \sqrt { 3 } + 3 i$ 
C)  $3 \sqrt { 3 } - 3 \mathrm { i }$ 
D)  $3 \sqrt { 3 } + 3 i$ 
A)  $- 3 \sqrt { 3 } - 3 i$ 
B)  $- 3 \sqrt { 3 } + 3 i$ 
C)  $3 \sqrt { 3 } - 3 \mathrm { i }$ 
D)  $3 \sqrt { 3 } + 3 i$

Question 8

Determine whether there is sufficient information for solving a triangle, with the given combination of angles and sides,
by the law of sines.
-B, a, and c

Accepted Answer

A)  Yes 
B)  No 
A)  Yes 
B)  No

Question 9

Plot the point.
-$$\left( 2 , \frac { 5 \pi } { 4 } \right)$$

Accepted Answer

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

Question 10

Find the quotient and write in rectangular form. First convert the numerator and denominator to trigonometric form.
-$$\frac { 8 + 2 i } { 3 - 9 i }$$

Accepted Answer

A)  $\frac { 1 } { 15 } + \frac { 13 } { 15 }$ i 
B)  $\frac { 14 } { 5 } + \frac { 22 } { 5 } i$ 
C)  $- \frac { 7 } { 12 } + \frac { 13 } { 72 } \mathrm { i }$ 
D)  $- \frac { 1 } { 72 }$ 
A)  $\frac { 1 } { 15 } + \frac { 13 } { 15 }$ i 
B)  $\frac { 14 } { 5 } + \frac { 22 } { 5 } i$ 
C)  $- \frac { 7 } { 12 } + \frac { 13 } { 72 } \mathrm { i }$ 
D)  $- \frac { 1 } { 72 }$

Question 11

Graph the complex number.
-$$4 \sqrt { 5 } - 4 i \sqrt { 5 }$$

Accepted Answer

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

Question 12

Determine two pairs of polar coordinates for the point with 0° ≤ θ < 360°.
-$$( - 2 \sqrt { 2 } , - 2 \sqrt { 2 } )$$

Accepted Answer

A)  $\left( 2 \sqrt { 2 } , 135 ^ { \circ } \right) , \left( - 2 \sqrt { 2 } , 225 ^ { \circ } \right)$ 
B)  $\left( 2 \sqrt { 2 } , 225 ^ { \circ } \right) , \left( - 2 \sqrt { 2 } , 45 ^ { \circ } \right)$ 
C)  $\left( 4,225 ^ { \circ } \right) , \left( - 4,45 ^ { \circ } \right)$ 
D)  $\left( 4,135 ^ { \circ } \right) , \left( - 4,225 ^ { \circ } \right)$ 
A)  $\left( 2 \sqrt { 2 } , 135 ^ { \circ } \right) , \left( - 2 \sqrt { 2 } , 225 ^ { \circ } \right)$ 
B)  $\left( 2 \sqrt { 2 } , 225 ^ { \circ } \right) , \left( - 2 \sqrt { 2 } , 45 ^ { \circ } \right)$ 
C)  $\left( 4,225 ^ { \circ } \right) , \left( - 4,45 ^ { \circ } \right)$ 
D)  $\left( 4,135 ^ { \circ } \right) , \left( - 4,225 ^ { \circ } \right)$

Question 13

Write the complex number in trigonometric form $$r ( \cos \theta + i \sin \theta )$$ with θ in the interval [0°, 360°).
-$8 \sqrt { 3 } + 8 \mathrm { i }$

Accepted Answer

A)  $16 \left( \cos 300 ^ { \circ } + i \sin 300 ^ { \circ } \right)$ 
B)  $16 \left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ } \right)$ 
C)  $16 \left( \cos 60 ^ { \circ } + i \sin 60 ^ { \circ } \right)$ 
D)  $16 \left( \cos 330 ^ { \circ } + i \sin 330 ^ { \circ } \right)$ 
A)  $16 \left( \cos 300 ^ { \circ } + i \sin 300 ^ { \circ } \right)$ 
B)  $16 \left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ } \right)$ 
C)  $16 \left( \cos 60 ^ { \circ } + i \sin 60 ^ { \circ } \right)$ 
D)  $16 \left( \cos 330 ^ { \circ } + i \sin 330 ^ { \circ } \right)$

Question 14

Graph the cycloid for t in the indicated interval.
-$$x = t - \sin t , y = 1 - \cos t , - 4 \pi \leq t \leq 4 \pi$$

Accepted Answer

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

Question 15

Find the product. Write the product in rectangular form, using exact values.
-[7( $\left. \left. \cos 45 ^ { \circ } + \mathrm { i } \sin 45 ^ { \circ } \right) \right] \left[ 2 \left( \cos 90 ^ { \circ } + \mathrm { i } \sin 90 ^ { \circ } \right) \right]$

Accepted Answer

A)  $- 14 \sqrt { 2 } - 14 \sqrt { 2 } \mathrm { i }$ 
B)  $7 \sqrt { 2 } - 28 \sqrt { 2 } \mathrm { i }$ 
C)  $2 \sqrt { 6 } - \sqrt { 2 } i$ 
D)  $- 7 \sqrt { 2 } + 7 \sqrt { 2 } i$ 
A)  $- 14 \sqrt { 2 } - 14 \sqrt { 2 } \mathrm { i }$ 
B)  $7 \sqrt { 2 } - 28 \sqrt { 2 } \mathrm { i }$ 
C)  $2 \sqrt { 6 } - \sqrt { 2 } i$ 
D)  $- 7 \sqrt { 2 } + 7 \sqrt { 2 } i$

Question 16

Determine whether the pair of vectors is orthogonal.
-$$\langle - 2,6 \rangle , \langle 6 , - 5 \rangle$$

Accepted Answer

A)  Yes 
B)  No 
A)  Yes 
B)  No

Question 17

Find the magnitude and direction angle (to the nearest tenth) for each vector. Give the measure of the direction angle as an angle in [0,360°].
-$\langle - 5 \sqrt { 3 } , 5 \rangle$

Accepted Answer

A)  $10 ; 330 ^ { \circ }$ 
B)  $20 ; 30 ^ { \circ }$ 
C)  $10 ; 150 ^ { \circ }$ 
D)  $20 ; 150 ^ { \circ }$ 
A)  $10 ; 330 ^ { \circ }$ 
B)  $20 ; 30 ^ { \circ }$ 
C)  $10 ; 150 ^ { \circ }$ 
D)  $20 ; 150 ^ { \circ }$

Question 18

Solve the problem.
-A projectile is fired with an initial velocity of 300 feet per second at an angle of $$70 ^ { \circ }$$ with the horizontal. In how many seconds will the projectile reach its maximum altitude? (Round your
Answer to the nearest tenth of a second.)

Accepted Answer

A)  5.4 sec 
B)  13.6 sec 
C)  16.9 sec 
D)  8.8 sec 
A)  5.4 sec 
B)  13.6 sec 
C)  16.9 sec 
D)  8.8 sec

Question 19

Find all solutions of the equation. Leave answers in trigonometric form.
-$$x ^ { 3 } - 8 = 0$$

Accepted Answer

A)  $\left\{ 2,2 \operatorname { cis } 60 ^ { \circ } , 2 \operatorname { cis } 240 ^ { \circ } \right\}$ 
B)  $\left\{ 2,2 \operatorname { cis } 60 ^ { \circ } , 2 \operatorname { cis } 300 ^ { \circ } \right\}$ 
C)  $\left\{ 2,2 \operatorname { cis } 300 ^ { \circ } , 2 \operatorname { cis } 300 ^ { \circ } \right\}$ 
D)  $\left\{ 2,2 \operatorname { cis } 120 ^ { \circ } , 2 \operatorname { cis } 240 ^ { \circ } \right\}$ 
A)  $\left\{ 2,2 \operatorname { cis } 60 ^ { \circ } , 2 \operatorname { cis } 240 ^ { \circ } \right\}$ 
B)  $\left\{ 2,2 \operatorname { cis } 60 ^ { \circ } , 2 \operatorname { cis } 300 ^ { \circ } \right\}$ 
C)  $\left\{ 2,2 \operatorname { cis } 300 ^ { \circ } , 2 \operatorname { cis } 300 ^ { \circ } \right\}$ 
D)  $\left\{ 2,2 \operatorname { cis } 120 ^ { \circ } , 2 \operatorname { cis } 240 ^ { \circ } \right\}$

Question 20

Solve the problem.
-Two tracking stations are on the equator 171 miles apart. A weather balloon is located on a bearing of $\mathrm { N } \mathrm {} 36 ^ { \circ } \mathrm { E }$ from the western station and a bearing of $\mathrm { N } \mathrm {} 18 ^ { \circ } \mathrm { E }$ from the eastern station. How far, to the nearest mile, is the balloon from the western station? Round to the nearest mile.

Accepted Answer

A)  $526 \mathrm { mi }$ 
B)  $448 \mathrm { mi }$ 
C)  $535 \mathrm { mi }$ 
D)  $439 \mathrm { mi }$ 
A)  $526 \mathrm { mi }$ 
B)  $448 \mathrm { mi }$ 
C)  $535 \mathrm { mi }$ 
D)  $439 \mathrm { mi }$

Solve the problem. -A force of $35 \mathrm { lb }$ is required to hold a $78 \mathrm { lb }$ toolbox on an incline. What angle does the incline make with the horizontal?

Find the polar coordinates of the point(s) of intersection of the given curves for 0 ≤ θ < 2π. - $\mathrm { r } = 4 \cos \theta , \mathrm { r } = - 1 + 2 \cos \theta$

Determine whether the pair of vectors is orthogonal. - $\langle 3,1 \rangle , \langle - 6,36 \rangle$

Find the quotient and write in rectangular form. First convert the numerator and denominator to trigonometric form. - $\frac { 5 \left( \cos 200 ^ { \circ } + i \sin 200 ^ { \circ } \right) } { 4 \left( \cos 50 ^ { \circ } + i \sin 50 ^ { \circ } \right) }$

Find the given power. Write answer in rectangular form. - $[ - 3 + 3 i \sqrt { 3 } ] ^ { 3 }$

Solve the problem. -Two forces of 575 newtons and 271 newtons act at a point. The resultant force is 657 newtons. Find the angle between the forces.

Write the complex number in rectangular form. - $6 \left( \cos 330 ^ { \circ } + i \sin 330 ^ { \circ } \right)$

Determine whether there is sufficient information for solving a triangle, with the given combination of angles and sides, by the law of sines. -B, a, and c

Plot the point. - $\left( 2 , \frac { 5 \pi } { 4 } \right)$ $Plot the point. - \left( 2 , \frac { 5 \pi } { 4 } \right)$

Find the quotient and write in rectangular form. First convert the numerator and denominator to trigonometric form. - $\frac { 8 + 2 i } { 3 - 9 i }$

Graph the complex number. - $4 \sqrt { 5 } - 4 i \sqrt { 5 }$ $Graph the complex number. - 4 \sqrt { 5 } - 4 i \sqrt { 5 }$

Determine two pairs of polar coordinates for the point with 0° ≤ θ < 360°. - $( - 2 \sqrt { 2 } , - 2 \sqrt { 2 } )$

Write the complex number in trigonometric form $r ( \cos \theta + i \sin \theta )$ with θ in the interval [0°, 360°). - $8 \sqrt { 3 } + 8 \mathrm { i }$

Graph the cycloid for t in the indicated interval. - $x = t - \sin t , y = 1 - \cos t , - 4 \pi \leq t \leq 4 \pi$ $Graph the cycloid for t in the indicated interval. - x = t - \sin t , y = 1 - \cos t , - 4 \pi \leq t \leq 4 \pi$

Find the product. Write the product in rectangular form, using exact values. -[7( $\left. \left. \cos 45 ^ { \circ } + \mathrm { i } \sin 45 ^ { \circ } \right) \right] \left[ 2 \left( \cos 90 ^ { \circ } + \mathrm { i } \sin 90 ^ { \circ } \right) \right]$

Determine whether the pair of vectors is orthogonal. - $\langle - 2,6 \rangle , \langle 6 , - 5 \rangle$

Find the magnitude and direction angle (to the nearest tenth) for each vector. Give the measure of the direction angle as an angle in [0,360°]. - $\langle - 5 \sqrt { 3 } , 5 \rangle$

Solve the problem. -A projectile is fired with an initial velocity of 300 feet per second at an angle of $70 ^ { \circ }$ with the horizontal. In how many seconds will the projectile reach its maximum altitude? (Round your Answer to the nearest tenth of a second.)

Find all solutions of the equation. Leave answers in trigonometric form. - $x ^ { 3 } - 8 = 0$

Equations and Inequalities

Graphs and Functions

Polynomial and Rational Functions

Inverse, Exponential, and Logarithmic Functions

Trigonometric Functions

The Circular Functions and Their Graphs

Trigonometric Identities and Equations

Systems and Matrices

Analytic Geometry

Further Topics in Algebra

Review of Basic Concepts

Filters

Exam 8: Applications of Trigonometry

Solve the problem. -A force of 35lb35 \mathrm { lb }35lb is required to hold a 78lb78 \mathrm { lb }78lb toolbox on an incline. What angle does the incline make with the horizontal?

Find the polar coordinates of the point(s) of intersection of the given curves for 0 ≤ θ < 2π. - r=4cos⁡θ,r=−1+2cos⁡θ\mathrm { r } = 4 \cos \theta , \mathrm { r } = - 1 + 2 \cos \thetar=4cosθ,r=−1+2cosθ

Determine whether the pair of vectors is orthogonal. - ⟨3,1⟩,⟨−6,36⟩\langle 3,1 \rangle , \langle - 6,36 \rangle⟨3,1⟩,⟨−6,36⟩

Find the given power. Write answer in rectangular form. - [−3+3i3]3[ - 3 + 3 i \sqrt { 3 } ] ^ { 3 }[−3+3i3​]3

Solve the problem. -Two forces of 575 newtons and 271 newtons act at a point. The resultant force is 657 newtons. Find the angle between the forces.

Write the complex number in rectangular form. - 6(cos⁡330∘+isin⁡330∘)6 \left( \cos 330 ^ { \circ } + i \sin 330 ^ { \circ } \right)6(cos330∘+isin330∘)

Determine whether there is sufficient information for solving a triangle, with the given combination of angles and sides, by the law of sines. -B, a, and c

Plot the point. - (2,5π4)\left( 2 , \frac { 5 \pi } { 4 } \right)(2,45π​)

Find the quotient and write in rectangular form. First convert the numerator and denominator to trigonometric form. - 8+2i3−9i\frac { 8 + 2 i } { 3 - 9 i }3−9i8+2i​

Graph the complex number. - 45−4i54 \sqrt { 5 } - 4 i \sqrt { 5 }45​−4i5​

Determine two pairs of polar coordinates for the point with 0° ≤ θ < 360°. - (−22,−22)( - 2 \sqrt { 2 } , - 2 \sqrt { 2 } )(−22​,−22​)

Write the complex number in trigonometric form r(cos⁡θ+isin⁡θ)r ( \cos \theta + i \sin \theta )r(cosθ+isinθ) with θ in the interval [0°, 360°). - 83+8i8 \sqrt { 3 } + 8 \mathrm { i }83​+8i

Graph the cycloid for t in the indicated interval. - x=t−sin⁡t,y=1−cos⁡t,−4π≤t≤4πx = t - \sin t , y = 1 - \cos t , - 4 \pi \leq t \leq 4 \pix=t−sint,y=1−cost,−4π≤t≤4π

Determine whether the pair of vectors is orthogonal. - ⟨−2,6⟩,⟨6,−5⟩\langle - 2,6 \rangle , \langle 6 , - 5 \rangle⟨−2,6⟩,⟨6,−5⟩

Find the magnitude and direction angle (to the nearest tenth) for each vector. Give the measure of the direction angle as an angle in [0,360°]. - ⟨−53,5⟩\langle - 5 \sqrt { 3 } , 5 \rangle⟨−53​,5⟩

Solve the problem. -A projectile is fired with an initial velocity of 300 feet per second at an angle of 70∘70 ^ { \circ }70∘ with the horizontal. In how many seconds will the projectile reach its maximum altitude? (Round your Answer to the nearest tenth of a second.)

Find all solutions of the equation. Leave answers in trigonometric form. - x3−8=0x ^ { 3 } - 8 = 0x3−8=0

Equations and Inequalities

Graphs and Functions

Polynomial and Rational Functions

Inverse, Exponential, and Logarithmic Functions

Trigonometric Functions

The Circular Functions and Their Graphs

Trigonometric Identities and Equations

Systems and Matrices

Analytic Geometry

Further Topics in Algebra

Review of Basic Concepts

Filters

Solve the problem. -A force of $35 \mathrm { lb }$ is required to hold a $78 \mathrm { lb }$ toolbox on an incline. What angle does the incline make with the horizontal?

Find the polar coordinates of the point(s) of intersection of the given curves for 0 ≤ θ < 2π. - $\mathrm { r } = 4 \cos \theta , \mathrm { r } = - 1 + 2 \cos \theta$

Determine whether the pair of vectors is orthogonal. - $\langle 3,1 \rangle , \langle - 6,36 \rangle$

Find the given power. Write answer in rectangular form. - $[ - 3 + 3 i \sqrt { 3 } ] ^ { 3 }$

Write the complex number in rectangular form. - $6 \left( \cos 330 ^ { \circ } + i \sin 330 ^ { \circ } \right)$

Plot the point. - $\left( 2 , \frac { 5 \pi } { 4 } \right)$ $Plot the point. - \left( 2 , \frac { 5 \pi } { 4 } \right)$

Find the quotient and write in rectangular form. First convert the numerator and denominator to trigonometric form. - $\frac { 8 + 2 i } { 3 - 9 i }$

Graph the complex number. - $4 \sqrt { 5 } - 4 i \sqrt { 5 }$ $Graph the complex number. - 4 \sqrt { 5 } - 4 i \sqrt { 5 }$

Determine two pairs of polar coordinates for the point with 0° ≤ θ < 360°. - $( - 2 \sqrt { 2 } , - 2 \sqrt { 2 } )$

Write the complex number in trigonometric form $r ( \cos \theta + i \sin \theta )$ with θ in the interval [0°, 360°). - $8 \sqrt { 3 } + 8 \mathrm { i }$

Graph the cycloid for t in the indicated interval. - $x = t - \sin t , y = 1 - \cos t , - 4 \pi \leq t \leq 4 \pi$ $Graph the cycloid for t in the indicated interval. - x = t - \sin t , y = 1 - \cos t , - 4 \pi \leq t \leq 4 \pi$

Determine whether the pair of vectors is orthogonal. - $\langle - 2,6 \rangle , \langle 6 , - 5 \rangle$

Find the magnitude and direction angle (to the nearest tenth) for each vector. Give the measure of the direction angle as an angle in [0,360°]. - $\langle - 5 \sqrt { 3 } , 5 \rangle$

Solve the problem. -A projectile is fired with an initial velocity of 300 feet per second at an angle of $70 ^ { \circ }$ with the horizontal. In how many seconds will the projectile reach its maximum altitude? (Round your Answer to the nearest tenth of a second.)

Find all solutions of the equation. Leave answers in trigonometric form. - $x ^ { 3 } - 8 = 0$