Question 1

Determine whether the Law of Sines or the Law of Cosine is needed to solve the triangle.Then solve the triangle. ​
A = 9, c = 6, B = 50°
​

Accepted Answer

A) Law of Sines; A ≈ 88.21°, C ≈ 41.79°, b ≈ 6.9 
B) Law of Cosines; A ≈ 88.21°, C ≈ 41.79°, b ≈ 6.9 
C) Law of Cosines; A ≈ 41.79°, C ≈ 88.21°, b ≈ 6.9 
D) Law of Sines; A ≈ 41.79°, C ≈ 88.21°, b ≈ 6.9 
E) Law of Sines; No solution 
A) Law of Sines; A ≈ 88.21°, C ≈ 41.79°, b ≈ 6.9 
B) Law of Cosines; A ≈ 88.21°, C ≈ 41.79°, b ≈ 6.9 
C) Law of Cosines; A ≈ 41.79°, C ≈ 88.21°, b ≈ 6.9 
D) Law of Sines; A ≈ 41.79°, C ≈ 88.21°, b ≈ 6.9 
E) Law of Sines; No solution

Question 2

Write the complex number shown below in trigonometric form.

Accepted Answer

A)  $6 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$ 
B)  $6 \left( \cos \frac { 5 \pi } { 3 } + i \sin \frac { 5 \pi } { 3 } \right)$ 
C)  $6 \left( \cos \frac { 11 \pi } { 6 } + i \sin \frac { 11 \pi } { 6 } \right)$ 
D)  $6 \left( \cos \frac { 4 \pi } { 3 } + i \sin \frac { 4 \pi } { 3 } \right)$ 
E)  $6 \left( \cos \frac { 5 \pi } { 4 } + i \sin \frac { 5 \pi } { 4 } \right)$ 
A)  $6 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$ 
B)  $6 \left( \cos \frac { 5 \pi } { 3 } + i \sin \frac { 5 \pi } { 3 } \right)$ 
C)  $6 \left( \cos \frac { 11 \pi } { 6 } + i \sin \frac { 11 \pi } { 6 } \right)$ 
D)  $6 \left( \cos \frac { 4 \pi } { 3 } + i \sin \frac { 4 \pi } { 3 } \right)$ 
E)  $6 \left( \cos \frac { 5 \pi } { 4 } + i \sin \frac { 5 \pi } { 4 } \right)$

Question 3

Find the projection of u onto v if , $\mathbf { u } = \langle 4,2 \rangle$ , $\mathbf { v } = \langle 5 , - 2 \rangle$ .

Accepted Answer

A)    $\left\langle \frac { 80 } { \sqrt { 29 } } , \frac { 32 } { \sqrt { 29 } } \right\rangle$ 
B)   $\left\langle \frac { 64 } { \sqrt { 29 } } , \frac { 80 } { \sqrt { 29 } } \right\rangle$ 
C)   $\left\langle \frac { 64 } { \sqrt { 29 } } , \frac { 32 } { \sqrt { 29 } } \right\rangle$ 
D)   $\left\langle \frac { 64 } { \sqrt { 29 } } , - \frac { 32 } { \sqrt { 29 } } \right\rangle$ 
E)   $\left\langle \frac { 80 } { \sqrt { 29 } } , - \frac { 32 } { \sqrt { 29 } } \right\rangle$ 
A)    $\left\langle \frac { 80 } { \sqrt { 29 } } , \frac { 32 } { \sqrt { 29 } } \right\rangle$ 
B)   $\left\langle \frac { 64 } { \sqrt { 29 } } , \frac { 80 } { \sqrt { 29 } } \right\rangle$ 
C)   $\left\langle \frac { 64 } { \sqrt { 29 } } , \frac { 32 } { \sqrt { 29 } } \right\rangle$ 
D)   $\left\langle \frac { 64 } { \sqrt { 29 } } , - \frac { 32 } { \sqrt { 29 } } \right\rangle$ 
E)   $\left\langle \frac { 80 } { \sqrt { 29 } } , - \frac { 32 } { \sqrt { 29 } } \right\rangle$

Question 4

Find the trigonometric form of the complex number shown below. $6 i$

Accepted Answer

A)  $6 ( \cos \pi + i \sin \pi )$ 
B)  $6 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$ 
C)  $6 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$ 
D)  $6 \left( \cos \frac { \pi } { 4 } + i \sin \frac { \pi } { 4 } \right)$ 
E)  $6 ( \cos 0 + i \sin 0 )$ 
A)  $6 ( \cos \pi + i \sin \pi )$ 
B)  $6 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$ 
C)  $6 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$ 
D)  $6 \left( \cos \frac { \pi } { 4 } + i \sin \frac { \pi } { 4 } \right)$ 
E)  $6 ( \cos 0 + i \sin 0 )$

Question 5

Find the component form of vector v with initial point (1, -7) and terminal point (-3, 1).

Accepted Answer

A)   $\mathbf { v } = \langle - 4 , - 8 \rangle$ 
B)  $\mathbf { v } = \langle - 4,8 \rangle$ 
C)   $v = \langle 8 , - 4 \rangle$ 
D)   $v = \langle 0 , - 4 \rangle$ 
E)   $\mathbf { v } = \langle - 8 , - 4 \rangle$ 
A)   $\mathbf { v } = \langle - 4 , - 8 \rangle$ 
B)  $\mathbf { v } = \langle - 4,8 \rangle$ 
C)   $v = \langle 8 , - 4 \rangle$ 
D)   $v = \langle 0 , - 4 \rangle$ 
E)   $\mathbf { v } = \langle - 8 , - 4 \rangle$

Question 6

The vector $\mathbf { u } = \langle 3900,4000 \rangle$ gives the number of units of two models of laptops produced by a company.The vector $\mathbf { u } = \langle 1450,1000 \rangle$ gives the prices (in dollars) of the two models of laptops, respectively.Identify the vector operation used to increase revenue by 6%.

Accepted Answer

A)  1.06(u + v) 
B) 1.06 $\left\| \mathbf { u } ^ { \prime } \mathbf { v } \right\|$ 
C)  $\mathbf { u } \cdot ( 1.06 ) \| \mathbf { v } \|$ 
D) 1.06 $\left( \mathbf { u } ^ { * } \mathbf { v } \right)$ 
E) 1.06 $\| \mathbf { u } \| ^ { . } \mathbf { v }$ 
A)  1.06(u + v) 
B) 1.06 $\left\| \mathbf { u } ^ { \prime } \mathbf { v } \right\|$ 
C)  $\mathbf { u } \cdot ( 1.06 ) \| \mathbf { v } \|$ 
D) 1.06 $\left( \mathbf { u } ^ { * } \mathbf { v } \right)$ 
E) 1.06 $\| \mathbf { u } \| ^ { . } \mathbf { v }$

Question 7

Perform the operation and leave the result in trigonometric form.   $\left( \cos 7 ^ { \circ } + i \sin 7 ^ { \circ } \right) \left( \cos 24 ^ { \circ } + i \sin 24 ^ { \circ } \right)$

Accepted Answer

A) cos 17° - isin 17° 
B) cos 31° - isin 31° 
C) cos 17° + isin 17° 
D) cos 31° + icos 31° 
E) cos 31° + isin 31° 
A) cos 17° - isin 17° 
B) cos 31° - isin 31° 
C) cos 17° + isin 17° 
D) cos 31° + icos 31° 
E) cos 31° + isin 31°

Question 8

A straight road makes an angle, A, of 10° with the horizontal.When the angle of elevation, B, of the sun is 56°, a vertical pole beside the road casts a shadow 8 feet long parallel to the road, see figure.Approximate the length of the pole.Round answer to two decimal places.

Accepted Answer

A) 2.48 feet 
B) 10.29 feet 
C) 6.22 feet 
D) 11.86 feet 
E) 6.94 feet 
A) 2.48 feet 
B) 10.29 feet 
C) 6.22 feet 
D) 11.86 feet 
E) 6.94 feet

Question 9

A traffic light weighing 16 pounds is suspended by two cables (see figure).Find the tension in each cable.Approximate the answer to one decimal place. ​
​   ​
Where b = 28°
​

Accepted Answer

A) 18.0 lb 
B) 17.0 lb 
C) 16.0 lb 
D) 19.0 lb 
E) 15.0 lb 
A) 18.0 lb 
B) 17.0 lb 
C) 16.0 lb 
D) 19.0 lb 
E) 15.0 lb

Question 10

Find the dot product of u and v.   $\begin{array} { l } 
\mathbf { u } = \langle 6,14 \rangle \
\mathbf { v } = \langle - 2,2 \rangle
\end{array}$

Accepted Answer

A) 18 
B) 14 
C) 20 
D) 12 
E) 16 
A) 18 
B) 14 
C) 20 
D) 12 
E) 16

Question 11

Perform the indicated operation in trigonometric form.Write the solution in standard form rounded to four decimal places. $\frac { 4 + i \sqrt { 3 } } { 8 - 3 i }$

Accepted Answer

A) 0.4682 + 0.2027i 
B) 0.4777 - 0.1791i 
C) 0.7197 + 0.6943i 
D) 0.5095 + 0.0254i 
E) 0.3672 + 0.3542i 
A) 0.4682 + 0.2027i 
B) 0.4777 - 0.1791i 
C) 0.7197 + 0.6943i 
D) 0.5095 + 0.0254i 
E) 0.3672 + 0.3542i

Question 12

Given a = 8, b = 14, and c = 12, use the Law of Cosines to solve the triangle for the value of C.Round your answer to two decimal places. ​

Accepted Answer

A) 60.33° 
B) 86.42° 
C) 58.81° 
D) 34.77° 
E) 80.44° 
A) 60.33° 
B) 86.42° 
C) 58.81° 
D) 34.77° 
E) 80.44°

Question 13

A vertical pole 29 feet tall stands on a hillside that makes an angle of 20° with the horizontal.Determine the approximate length of cable that would be needed to reach from the top of the pole to a point 65 feet downhill from the base of the pole.Round your answer to two decimal places. ​

Accepted Answer

A) 70.59 feet 
B) 61.45 feet 
C) 84.70 feet 
D) 75.29 feet 
E) 79.72 feet 
A) 70.59 feet 
B) 61.45 feet 
C) 84.70 feet 
D) 75.29 feet 
E) 79.72 feet

Question 14

Use the Heron's formula to find the area of the triangle.Round your answer to two decimal places. ​
A = 10, b = 14, c = 19
​

Accepted Answer

A) 14.68 
B) 84.31 
C) 54.99 
D) 31.75 
E) 68.09 
A) 14.68 
B) 84.31 
C) 54.99 
D) 31.75 
E) 68.09

Question 15

In the figure, $\alpha$  and $\beta$ are positive angles.      
A = 3, b = 12
Use $\alpha$  = arcsin(0.25sin$\beta$) to write c as a function of $\beta$.

Accepted Answer

A)  $c = \frac { 12 \sin [ \pi - \beta - \arcsin ( 0.25 \beta ) ] } { \sin \beta }$ 
B)  $c = \frac { 12 \sin [ \pi + \beta - \arcsin ( 0.25 \beta ) ] } { \sin \beta }$ 
C)  $c = \frac { 0.25 \sin [ \pi - \beta - \arcsin ( 12 \beta ) ] } { \sin \beta }$ 
D)  $c = \frac { 12 \sin [ \pi + \beta + \arcsin ( 0.25 \beta ) ] } { \sin \beta }$ 
E)  $c = \frac { 12 \sin [ \pi - \beta + \arcsin ( 0.25 \beta ) ] } { \sin \beta }$ 
A)  $c = \frac { 12 \sin [ \pi - \beta - \arcsin ( 0.25 \beta ) ] } { \sin \beta }$ 
B)  $c = \frac { 12 \sin [ \pi + \beta - \arcsin ( 0.25 \beta ) ] } { \sin \beta }$ 
C)  $c = \frac { 0.25 \sin [ \pi - \beta - \arcsin ( 12 \beta ) ] } { \sin \beta }$ 
D)  $c = \frac { 12 \sin [ \pi + \beta + \arcsin ( 0.25 \beta ) ] } { \sin \beta }$ 
E)  $c = \frac { 12 \sin [ \pi - \beta + \arcsin ( 0.25 \beta ) ] } { \sin \beta }$

Question 16

Find the angle $\theta$  between the vectors.  
  $\begin{array} { l } 
\mathbf { u } = - 7 \mathbf { i } - 8 \mathbf { j } \
\mathbf { v } = - 9 \mathbf { i } + 2 \mathbf { j }
\end{array}$  
(Round the answer to 2 decimal places.)

Accepted Answer

A)  $71.34 ^ { \circ }$ 
B)  $66.34 ^ { \circ }$ 
C)  $61.34 ^ { \circ }$ 
D)  $81.34 ^ { \circ }$ 
E)  $76.34 ^ { \circ }$ 
A)  $71.34 ^ { \circ }$ 
B)  $66.34 ^ { \circ }$ 
C)  $61.34 ^ { \circ }$ 
D)  $81.34 ^ { \circ }$ 
E)  $76.34 ^ { \circ }$

Question 17

A park ranger at point A observes a fire in the direction N25°26'E.Another ranger at point B, 4 miles due east of A, sites the same fire at N58°34'W.Determine the distance from point B to the fire.Round answer to two decimal places.

Accepted Answer

A) 1.73 miles 
B) 3.63 miles 
C) 2.05 miles 
D) 4.08 miles 
E) 3.43 miles 
A) 1.73 miles 
B) 3.63 miles 
C) 2.05 miles 
D) 4.08 miles 
E) 3.43 miles

Question 18

Plot the complex number and find its absolute value.  
-3 + 9i

Accepted Answer

A)      $| - 3 + 9 i | = \sqrt { 76 }$ 
B)       $| - 3 + 9 i | = \sqrt { 12 }$ 
C)       $| - 3 + 9 i | = \sqrt { 90 }$ 
D)       $| - 3 + 9 i | = \sqrt { 2 }$ 
E)       $| - 3 + 9 i | = \sqrt { 86 }$ 
A)      $| - 3 + 9 i | = \sqrt { 76 }$ 
B)       $| - 3 + 9 i | = \sqrt { 12 }$ 
C)       $| - 3 + 9 i | = \sqrt { 90 }$ 
D)       $| - 3 + 9 i | = \sqrt { 2 }$ 
E)       $| - 3 + 9 i | = \sqrt { 86 }$

Question 19

Represent the complex number below graphically. -3+ 3i

Accepted Answer

A)    
B) ​   
C) ​   
D)    
E) ​   
A)    
B) ​   
C) ​   
D)    
E) ​

Question 20

Find the magnitude and direction angle of $\mathbf { v } = - 2 \mathbf { i } - 6 \mathbf { j }$ .Round the direction angle to the nearest degree.

Accepted Answer

A)   ||v|| = $3 \sqrt { 7 }$ ; $\theta$ = 269° 
B)  ||v|| = $\sqrt { 30 }$ ; $\theta$ = 239° 
C)  ||v|| = $2 \sqrt { 13 }$ ; $\theta$ = 260° 
D)  ||v|| = $2 \sqrt { 10 }$ ; $\theta$ = 252° 
E)  ||v|| = $\sqrt { 67 }$ ; $\theta$= 279° 
A)   ||v|| = $3 \sqrt { 7 }$ ; $\theta$ = 269° 
B)  ||v|| = $\sqrt { 30 }$ ; $\theta$ = 239° 
C)  ||v|| = $2 \sqrt { 13 }$ ; $\theta$ = 260° 
D)  ||v|| = $2 \sqrt { 10 }$ ; $\theta$ = 252° 
E)  ||v|| = $\sqrt { 67 }$ ; $\theta$= 279°

Determine whether the Law of Sines or the Law of Cosine is needed to solve the triangle.Then solve the triangle. A = 9, c = 6, B = 50°

Write the complex number shown below in trigonometric form.

Find the projection of u onto v if , $\mathbf { u } = \langle 4,2 \rangle$ , $\mathbf { v } = \langle 5 , - 2 \rangle$ .

Find the trigonometric form of the complex number shown below. $6 i$

Find the component form of vector v with initial point (1, -7) and terminal point (-3, 1).

Perform the operation and leave the result in trigonometric form. $\left( \cos 7 ^ { \circ } + i \sin 7 ^ { \circ } \right) \left( \cos 24 ^ { \circ } + i \sin 24 ^ { \circ } \right)$

A straight road makes an angle, A, of 10° with the horizontal.When the angle of elevation, B, of the sun is 56°, a vertical pole beside the road casts a shadow 8 feet long parallel to the road, see figure.Approximate the length of the pole.Round answer to two decimal places.

A traffic light weighing 16 pounds is suspended by two cables (see figure).Find the tension in each cable.Approximate the answer to one decimal place. Where b = 28°

Find the dot product of u and v. =\langle6,14\rangle =\langle-2,2\rangle

Perform the indicated operation in trigonometric form.Write the solution in standard form rounded to four decimal places. $\frac { 4 + i \sqrt { 3 } } { 8 - 3 i }$

Given a = 8, b = 14, and c = 12, use the Law of Cosines to solve the triangle for the value of C.Round your answer to two decimal places.

A vertical pole 29 feet tall stands on a hillside that makes an angle of 20° with the horizontal.Determine the approximate length of cable that would be needed to reach from the top of the pole to a point 65 feet downhill from the base of the pole.Round your answer to two decimal places.

Use the Heron's formula to find the area of the triangle.Round your answer to two decimal places. A = 10, b = 14, c = 19

In the figure, $\alpha$ and $\beta$ are positive angles. $In the figure, \alpha and \beta are positive angles. A = 3, b = 12 Use \alpha = arcsin(0.25sin \beta ) to write c as a function of \beta .$ A = 3, b = 12 Use $\alpha$ = arcsin(0.25sin $\beta$ ) to write c as a function of $\beta$ .

Find the angle $\theta$ between the vectors. =-7-8 =-9+2 (Round the answer to 2 decimal places.)

A park ranger at point A observes a fire in the direction N25°26'E.Another ranger at point B, 4 miles due east of A, sites the same fire at N58°34'W.Determine the distance from point B to the fire.Round answer to two decimal places.

Plot the complex number and find its absolute value. -3 + 9i

Represent the complex number below graphically. -3+ 3i

Find the magnitude and direction angle of $\mathbf { v } = - 2 \mathbf { i } - 6 \mathbf { j }$ .Round the direction angle to the nearest degree.

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometry

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Systems Of Equations and Inequalities

Matrices and Determinants

Sequences Series and Probability

Topics In Analytic Geometry

Analytic Geometry In Three Dimensions

Limits and An Introduction To Calculus

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Exam 6: Additional Topics In Trigonometery

Determine whether the Law of Sines or the Law of Cosine is needed to solve the triangle.Then solve the triangle. ​ A = 9, c = 6, B = 50° ​

Write the complex number shown below in trigonometric form.

Find the projection of u onto v if , u=⟨4,2⟩\mathbf { u } = \langle 4,2 \rangleu=⟨4,2⟩ , v=⟨5,−2⟩\mathbf { v } = \langle 5 , - 2 \ranglev=⟨5,−2⟩ .

Find the trigonometric form of the complex number shown below. 6i6 i6i

Find the component form of vector v with initial point (1, -7) and terminal point (-3, 1).

Perform the operation and leave the result in trigonometric form. (cos⁡7∘+isin⁡7∘)(cos⁡24∘+isin⁡24∘)\left( \cos 7 ^ { \circ } + i \sin 7 ^ { \circ } \right) \left( \cos 24 ^ { \circ } + i \sin 24 ^ { \circ } \right)(cos7∘+isin7∘)(cos24∘+isin24∘)

A straight road makes an angle, A, of 10° with the horizontal.When the angle of elevation, B, of the sun is 56°, a vertical pole beside the road casts a shadow 8 feet long parallel to the road, see figure.Approximate the length of the pole.Round answer to two decimal places.

A traffic light weighing 16 pounds is suspended by two cables (see figure).Find the tension in each cable.Approximate the answer to one decimal place. ​ ​ ​ Where b = 28° ​

Find the dot product of u and v. =\langle6,14\rangle =\langle-2,2\rangle

Perform the indicated operation in trigonometric form.Write the solution in standard form rounded to four decimal places. 4+i38−3i\frac { 4 + i \sqrt { 3 } } { 8 - 3 i }8−3i4+i3​​

Given a = 8, b = 14, and c = 12, use the Law of Cosines to solve the triangle for the value of C.Round your answer to two decimal places. ​

A vertical pole 29 feet tall stands on a hillside that makes an angle of 20° with the horizontal.Determine the approximate length of cable that would be needed to reach from the top of the pole to a point 65 feet downhill from the base of the pole.Round your answer to two decimal places. ​

Use the Heron's formula to find the area of the triangle.Round your answer to two decimal places. ​ A = 10, b = 14, c = 19 ​

In the figure, α\alphaα and β\betaβ are positive angles. A = 3, b = 12 Use α\alphaα = arcsin(0.25sin β\betaβ ) to write c as a function of β\betaβ .

Find the angle θ\thetaθ between the vectors. =-7-8 =-9+2 (Round the answer to 2 decimal places.)

A park ranger at point A observes a fire in the direction N25°26'E.Another ranger at point B, 4 miles due east of A, sites the same fire at N58°34'W.Determine the distance from point B to the fire.Round answer to two decimal places.

Plot the complex number and find its absolute value. -3 + 9i

Represent the complex number below graphically. -3+ 3i

Find the magnitude and direction angle of v=−2i−6j\mathbf { v } = - 2 \mathbf { i } - 6 \mathbf { j }v=−2i−6j .Round the direction angle to the nearest degree.

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometry

Analytic Trigonometry

Systems Of Equations and Inequalities

Matrices and Determinants

Sequences Series and Probability

Topics In Analytic Geometry

Analytic Geometry In Three Dimensions

Limits and An Introduction To Calculus

Filters

Determine whether the Law of Sines or the Law of Cosine is needed to solve the triangle.Then solve the triangle. A = 9, c = 6, B = 50°

Find the projection of u onto v if , $\mathbf { u } = \langle 4,2 \rangle$ , $\mathbf { v } = \langle 5 , - 2 \rangle$ .

Find the trigonometric form of the complex number shown below. $6 i$

Perform the operation and leave the result in trigonometric form. $\left( \cos 7 ^ { \circ } + i \sin 7 ^ { \circ } \right) \left( \cos 24 ^ { \circ } + i \sin 24 ^ { \circ } \right)$

A traffic light weighing 16 pounds is suspended by two cables (see figure).Find the tension in each cable.Approximate the answer to one decimal place. Where b = 28°

Perform the indicated operation in trigonometric form.Write the solution in standard form rounded to four decimal places. $\frac { 4 + i \sqrt { 3 } } { 8 - 3 i }$

Given a = 8, b = 14, and c = 12, use the Law of Cosines to solve the triangle for the value of C.Round your answer to two decimal places.

A vertical pole 29 feet tall stands on a hillside that makes an angle of 20° with the horizontal.Determine the approximate length of cable that would be needed to reach from the top of the pole to a point 65 feet downhill from the base of the pole.Round your answer to two decimal places.

Use the Heron's formula to find the area of the triangle.Round your answer to two decimal places. A = 10, b = 14, c = 19

In the figure, $\alpha$ and $\beta$ are positive angles. $In the figure, \alpha and \beta are positive angles. A = 3, b = 12 Use \alpha = arcsin(0.25sin \beta ) to write c as a function of \beta .$ A = 3, b = 12 Use $\alpha$ = arcsin(0.25sin $\beta$ ) to write c as a function of $\beta$ .

Find the angle $\theta$ between the vectors. =-7-8 =-9+2 (Round the answer to 2 decimal places.)

Find the magnitude and direction angle of $\mathbf { v } = - 2 \mathbf { i } - 6 \mathbf { j }$ .Round the direction angle to the nearest degree.