Question 1

Use the Law of Sines to solve for C and B.Round your answer to two decimal places. ​
A = 60°, a = 36, c = 40
​

Accepted Answer

A) C ≈ 105.79°, B ≈ 45.79°or​C ≈ 74.21°, B ≈ 14.21° 
B) ​C ≈ 74.21°, B ≈ 45.79°or​C ≈ 105.79°, B ≈ 14.21° 
C) ​C ≈ 10°, B ≈ 110°or​C ≈ 25.5°, B ≈ 104.5° 
D) ​C ≈ 45.79°, B ≈ 74.21°or​C ≈ 14.21°, B ≈ 105.79° 
E) ​C ≈ 60°, B ≈ 60°or​C ≈ 45°, B ≈ 75°

Question 2

Use the law of Cosines to solve the given triangle.Round your answer to two decimal places. ​
A = 14, b = 7, C = 118°
​  ​

Accepted Answer

A) A ≈ 42.33°, B ≈ 19.67°, c ≈ 18.36 
B) A ≈ 17.67°, B ≈ 17.67°, c ≈ 7 
C) A ≈ 44.33°, B ≈ 44.33°, c ≈ 7 
D) A ≈ 17.67°, B ≈ 18.36°, c ≈ 18.36 
E) A ≈ 44.33°, B ≈ 17.67°, c ≈ 7

Question 3

Find values for b such that the triangle has one solution.  
A = 62°, a = 304.6

Accepted Answer

A)  $b \geq 304.6 , b = \frac { 304.6 } { \sin 62 ^ { \circ } }$ 
B)  $b \leq 304.6 , b \neq \frac { 304.6 } { \sin 62 ^ { \circ } }$ 
C)  $b > 304.6 , b > \frac { 304.6 } { \sin 62 ^ { \circ } }$ 
D)  $b \leq 304.6 , b = \frac { 304.6 } { \sin 62 ^ { \circ } }$ 
E)  $b \geq 304.6 , b \geq \frac { 304.6 } { \sin 62 ^ { \circ } }$

Question 4

Determine whether u are v and orthogonal, parallel, or neither. $\mathbf { u } = \left\langle \frac { - 4 } { 3 } , \frac { 3 } { 2 } \right\rangle , \mathbf { v } = \langle 16 , - 18 \rangle$

Accepted Answer

A) neither 
B) parallel 
C) orthogonal

Question 5

Represent the following complex number graphically.  
  $$3 \left( \cos 160 ^ { \circ } + i \sin 160 ^ { \circ } \right)$$

Accepted Answer

A)   
B)   
C)    
D)   
E)

Question 6

Determine whether the Law of Sines or the Law of Cosines is needed to solve the triangle.Then solve the triangle.Round your answer to two decimal places. ​
A = 46°, B = 39°, c = 1.4
​

Accepted Answer

A) Law of Cosines; No solution 
B) Law of Sines; No solution 
C) Law of Cosines; C = 95°, a ≈ 0.88, b ≈ 1.01 
D) Law of Sines; C = 95°, a ≈ 1.01, b ≈ 0.88 
E) Law of Sines; C = 95°, a ≈ 1.01, b ≈ 1.01

Question 7

Find the vector v that has a magnitude of 9 and is in the same direction as u, where $\mathbf { u } = \langle 6 , - 5 \rangle$ .

Accepted Answer

A)   $\mathbf { v } = \left\langle \frac { 5 \sqrt { 61 } } { 61 } , - \frac { 6 \sqrt { 61 } } { 61 } \right\rangle$ 
B)   $\mathrm { v } = \left\langle \frac { 45 \sqrt { 61 } } { 61 } , - \frac { 54 \sqrt { 61 } } { 61 } \right\rangle$ 
C)   $\mathbf { v } = \left\langle \frac { \sqrt { 61 } } { 61 } , - \frac { \sqrt { 61 } } { 61 } \right\rangle$ 
D)   $\mathbf { v } = \left\langle \frac { 6 \sqrt { 61 } } { 61 } , - \frac { 5 \sqrt { 61 } } { 61 } \right\rangle$ 
E)   $\mathbf { v } = \left\langle \frac { 54 \sqrt { 61 } } { 61 } , - \frac { 45 \sqrt { 61 } } { 61 } \right\rangle$

Question 8

Determine the angle $\theta$ in the design of the streetlight shown in the following figure.Round your answer upto decimal place.   $$a = 6 , b = 7 \frac { 1 } { 2 } , c = 5$$

Accepted Answer

A) 80.5° 
B) 95.5° 
C) 75.5° 
D) 85.5° 
E) 90.5°

Question 9

Use the Law of Sines to solve (if possible) the triangle.Round your answers to two decimal places. ​
A = 120°, a = b = 36
​

Accepted Answer

A) B ≈ 108°, C ≈ 36°, c ≈ 44 
B) B ≈ 36°, C ≈ 108°, c ≈ 23.5 
C) B ≈ 105°, C ≈ 39°, c ≈ 23.5 
D) B ≈ 29°, C ≈ 115°, c ≈ 23.5 
E) No Solution

Question 10

Determine the area of a triangle having the following measurements.Round your answer to two decimal places. B = 64 $^\circ$ 31 $'$ , a = 10  and c = 8.

Accepted Answer

A) 43.33 sq. units 
B) 32.50 sq. units 
C) 28.89 sq. units 
D) 36.11 sq. units 
E) 39.72 sq. units

Question 11

Find a unit vector in the direction of the given vector.  
  $\mathbf { u } = \langle 4,0 \rangle$

Accepted Answer

A)  $\langle 1,1 \rangle$ 
B)  $\langle 1,0 \rangle$ 
C)  $\langle 0,0 \rangle$ 
D)  $\langle 4,0 \rangle$ 
E)  $\langle 0,1 \rangle$

Question 12

Find the angle $\theta$  between the vectors.   $\begin{array} { l } 
\mathbf { u } = \langle 6,2 \rangle \
\mathbf { v } = \langle 7,0 \rangle
\end{array}$  
(Round the answer to 2 decimal places.)

Accepted Answer

A)  $18.43 ^ { \circ }$ 
B)  $33.43 ^ { \circ }$ 
C)  $23.43 ^ { \circ }$ 
D)  $28.43 ^ { \circ }$ 
E)  $38.43 ^ { \circ }$

Question 13

Given $\mathbf { u } = \langle - 5 , - 6 \rangle$ and $\mathbf { v } = \langle - 6 , - 5 \rangle$ , find $u^. v $ .

Accepted Answer

A) 0 
B) 30 
C) 60 
D) 61 
E) -11

Question 14

Use the vectors $\mathbf { u } = \langle 3,6 \rangle$ , $\mathbf { v } = \langle - 6,5 \rangle$ to find the indicated quantity.State whether the result is a vector or a scalar.   $( \mathbf { u } \cdot \mathbf { v } ) \mathbf { v }$

Accepted Answer

A)  $\langle - 72,60 \rangle$ ; vector 
B)  $\langle - 72,64 \rangle$ ; vector 
C) -72; scalar 
D) 60; scalar 
E)  $\langle - 72,58 \rangle$ ; vector

Question 15

A force of F pounds is required to pull an object weighing W pounds up a ramp inclined at θ degrees from the horizontal. ​
Find F if W = 100 pounds and θ = 11°.Approximate the answer to one decimal place.
​

Accepted Answer

A) 18.1 lb 
B) 21.1 lb 
C) 19.1 lb 
D) 20.1 lb 
E) 17.1 lb

Question 16

Write the complex number in trigonometric form.

Accepted Answer

A)  $Z = 3 ( \cos \pi - i \cos \pi )$ 
B)   $Z = 3 ( \cos \pi + i \sin \pi )$ 
C)  $Z = - 3 ( \cos \pi + i \sin \pi )$ 
D)   $Z = 3 \left( \cos \frac { \pi } { 2 } - i \sin \frac { \pi } { 2 } \right)$ 
E)   $Z = 3 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$

Question 17

Using the figure below, sketch a graph of the given vector.[The graphs in the answer choices are drawn to the same scale as the graph below.] 
 -u

Accepted Answer

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

Question 18

Because of prevailing winds, a tree grew so that it was leaning 6° from the vertical.At a point 43 meters from the tree, the angle of elevation to the top of the tree is 30° (see figure).Find the height a of the tree.    c = 43 m
B = 96°

(Round your answer to two decimal places.)

Accepted Answer

A) 24.58 m 
B) 26.58 m 
C) 25.58 m 
D) 28.58 m 
E) 27.58 m

Question 19

To determine the distance between two aircraft, a tracking station continuously determines the distance to each aircraft and the angle A between them (see figure).Determine the distance a between the planes when A = 44°, b = 37 miles, and c = 22 miles. ​  ​

Accepted Answer

A) 35.60 miles 
B) 31.81 miles 
C) 26.11 miles 
D) 11.27 miles 
E) 54.99 miles

Question 20

Use the vectors $\mathbf { u } = \langle 2,2 \rangle$ , $\mathbf { v } = \langle - 2,2 \rangle$ to find the indicated quantity.State whether the result is a vector or a scalar.   $3 \mathbf { u } \cdot \mathbf { v }$

Accepted Answer

A)  $\langle 0,2 \rangle$ ; vector 
B) -2; scalar 
C) 2; scalar 
D)  $\langle 4 , - 4 \rangle$ ; vector 
E) 0; scalar

Exam 6: Additional Topics In Trigonometery

Use the Law of Sines to solve for C and B.Round your answer to two decimal places. A = 60°, a = 36, c = 40

Use the law of Cosines to solve the given triangle.Round your answer to two decimal places. A = 14, b = 7, C = 118°

Find values for b such that the triangle has one solution. A = 62°, a = 304.6

Determine whether u are v and orthogonal, parallel, or neither. $\mathbf { u } = \left\langle \frac { - 4 } { 3 } , \frac { 3 } { 2 } \right\rangle , \mathbf { v } = \langle 16 , - 18 \rangle$

Represent the following complex number graphically. $3 \left( \cos 160 ^ { \circ } + i \sin 160 ^ { \circ } \right)$ $Represent the following complex number graphically. 3 \left( \cos 160 ^ { \circ } + i \sin 160 ^ { \circ } \right) A) B) C) D) E)$

Determine whether the Law of Sines or the Law of Cosines is needed to solve the triangle.Then solve the triangle.Round your answer to two decimal places. A = 46°, B = 39°, c = 1.4

Find the vector v that has a magnitude of 9 and is in the same direction as u, where $\mathbf { u } = \langle 6 , - 5 \rangle$ .

Use the Law of Sines to solve (if possible) the triangle.Round your answers to two decimal places. A = 120°, a = b = 36

Determine the area of a triangle having the following measurements.Round your answer to two decimal places. B = 64 $^\circ$ 31 $'$ , a = 10 and c = 8.

Find a unit vector in the direction of the given vector. $\mathbf { u } = \langle 4,0 \rangle$

Find the angle $\theta$ between the vectors. $\begin{array} { l } \mathbf { u } = \langle 6,2 \rangle \\\mathbf { v } = \langle 7,0 \rangle\end{array}$ (Round the answer to 2 decimal places.)

Given $\mathbf { u } = \langle - 5 , - 6 \rangle$ and $\mathbf { v } = \langle - 6 , - 5 \rangle$ , find $u^. v$ .

Use the vectors $\mathbf { u } = \langle 3,6 \rangle$ , $\mathbf { v } = \langle - 6,5 \rangle$ to find the indicated quantity.State whether the result is a vector or a scalar. $( \mathbf { u } \cdot \mathbf { v } ) \mathbf { v }$

A force of F pounds is required to pull an object weighing W pounds up a ramp inclined at θ degrees from the horizontal. Find F if W = 100 pounds and θ = 11°.Approximate the answer to one decimal place.

Using the figure below, sketch a graph of the given vector.[The graphs in the answer choices are drawn to the same scale as the graph below.] -u

Because of prevailing winds, a tree grew so that it was leaning 6° from the vertical.At a point 43 meters from the tree, the angle of elevation to the top of the tree is 30° (see figure) .Find the height a of the tree. c = 43 m B = 96° (Round your answer to two decimal places.)

To determine the distance between two aircraft, a tracking station continuously determines the distance to each aircraft and the angle A between them (see figure) .Determine the distance a between the planes when A = 44°, b = 37 miles, and c = 22 miles.

Use the vectors $\mathbf { u } = \langle 2,2 \rangle$ , $\mathbf { v } = \langle - 2,2 \rangle$ to find the indicated quantity.State whether the result is a vector or a scalar. $3 \mathbf { u } \cdot \mathbf { v }$

Exam 6: Additional Topics In Trigonometery

Use the Law of Sines to solve for C and B.Round your answer to two decimal places. ​ A = 60°, a = 36, c = 40 ​

Use the law of Cosines to solve the given triangle.Round your answer to two decimal places. ​ A = 14, b = 7, C = 118° ​ ​

Find values for b such that the triangle has one solution. A = 62°, a = 304.6

Determine whether u are v and orthogonal, parallel, or neither. u=⟨−43,32⟩,v=⟨16,−18⟩\mathbf { u } = \left\langle \frac { - 4 } { 3 } , \frac { 3 } { 2 } \right\rangle , \mathbf { v } = \langle 16 , - 18 \rangleu=⟨3−4​,23​⟩,v=⟨16,−18⟩

Represent the following complex number graphically. 3(cos⁡160∘+isin⁡160∘) 3 \left( \cos 160 ^ { \circ } + i \sin 160 ^ { \circ } \right) 3(cos160∘+isin160∘)

Determine whether the Law of Sines or the Law of Cosines is needed to solve the triangle.Then solve the triangle.Round your answer to two decimal places. ​ A = 46°, B = 39°, c = 1.4 ​

Find the vector v that has a magnitude of 9 and is in the same direction as u, where u=⟨6,−5⟩\mathbf { u } = \langle 6 , - 5 \rangleu=⟨6,−5⟩ .

Determine the angle θ\thetaθ in the design of the streetlight shown in the following figure.Round your answer upto decimal place. a=6,b=712,c=5a = 6 , b = 7 \frac { 1 } { 2 } , c = 5a=6,b=721​,c=5

Use the Law of Sines to solve (if possible) the triangle.Round your answers to two decimal places. ​ A = 120°, a = b = 36 ​

Determine the area of a triangle having the following measurements.Round your answer to two decimal places. B = 64 ∘^\circ∘ 31 ′'′ , a = 10 and c = 8.

Find a unit vector in the direction of the given vector. u=⟨4,0⟩\mathbf { u } = \langle 4,0 \rangleu=⟨4,0⟩

Find the angle θ\thetaθ between the vectors. u=⟨6,2⟩v=⟨7,0⟩\begin{array} { l } \mathbf { u } = \langle 6,2 \rangle \\\mathbf { v } = \langle 7,0 \rangle\end{array}u=⟨6,2⟩v=⟨7,0⟩​ (Round the answer to 2 decimal places.)

Given u=⟨−5,−6⟩\mathbf { u } = \langle - 5 , - 6 \rangleu=⟨−5,−6⟩ and v=⟨−6,−5⟩\mathbf { v } = \langle - 6 , - 5 \ranglev=⟨−6,−5⟩ , find u.vu^. v u.v .

Use the vectors u=⟨3,6⟩\mathbf { u } = \langle 3,6 \rangleu=⟨3,6⟩ , v=⟨−6,5⟩\mathbf { v } = \langle - 6,5 \ranglev=⟨−6,5⟩ to find the indicated quantity.State whether the result is a vector or a scalar. (u⋅v) v( \mathbf { u } \cdot \mathbf { v } ) \mathbf { v }(u⋅v) v

A force of F pounds is required to pull an object weighing W pounds up a ramp inclined at θ degrees from the horizontal. ​ Find F if W = 100 pounds and θ = 11°.Approximate the answer to one decimal place. ​

Write the complex number in trigonometric form.

Using the figure below, sketch a graph of the given vector.[The graphs in the answer choices are drawn to the same scale as the graph below.] -u

Because of prevailing winds, a tree grew so that it was leaning 6° from the vertical.At a point 43 meters from the tree, the angle of elevation to the top of the tree is 30° (see figure) .Find the height a of the tree. c = 43 m B = 96° (Round your answer to two decimal places.)

To determine the distance between two aircraft, a tracking station continuously determines the distance to each aircraft and the angle A between them (see figure) .Determine the distance a between the planes when A = 44°, b = 37 miles, and c = 22 miles. ​ ​

Use the vectors u=⟨2,2⟩\mathbf { u } = \langle 2,2 \rangleu=⟨2,2⟩ , v=⟨−2,2⟩\mathbf { v } = \langle - 2,2 \ranglev=⟨−2,2⟩ to find the indicated quantity.State whether the result is a vector or a scalar. 3u⋅v3 \mathbf { u } \cdot \mathbf { v }3u⋅v

Use the Law of Sines to solve for C and B.Round your answer to two decimal places. A = 60°, a = 36, c = 40

Use the law of Cosines to solve the given triangle.Round your answer to two decimal places. A = 14, b = 7, C = 118°

Determine whether u are v and orthogonal, parallel, or neither. $\mathbf { u } = \left\langle \frac { - 4 } { 3 } , \frac { 3 } { 2 } \right\rangle , \mathbf { v } = \langle 16 , - 18 \rangle$

Represent the following complex number graphically. $3 \left( \cos 160 ^ { \circ } + i \sin 160 ^ { \circ } \right)$ $Represent the following complex number graphically. 3 \left( \cos 160 ^ { \circ } + i \sin 160 ^ { \circ } \right) A) B) C) D) E)$

Determine whether the Law of Sines or the Law of Cosines is needed to solve the triangle.Then solve the triangle.Round your answer to two decimal places. A = 46°, B = 39°, c = 1.4

Find the vector v that has a magnitude of 9 and is in the same direction as u, where $\mathbf { u } = \langle 6 , - 5 \rangle$ .

Use the Law of Sines to solve (if possible) the triangle.Round your answers to two decimal places. A = 120°, a = b = 36

Determine the area of a triangle having the following measurements.Round your answer to two decimal places. B = 64 $^\circ$ 31 $'$ , a = 10 and c = 8.

Find a unit vector in the direction of the given vector. $\mathbf { u } = \langle 4,0 \rangle$

Find the angle $\theta$ between the vectors. $\begin{array} { l } \mathbf { u } = \langle 6,2 \rangle \\\mathbf { v } = \langle 7,0 \rangle\end{array}$ (Round the answer to 2 decimal places.)

Given $\mathbf { u } = \langle - 5 , - 6 \rangle$ and $\mathbf { v } = \langle - 6 , - 5 \rangle$ , find $u^. v$ .

Use the vectors $\mathbf { u } = \langle 3,6 \rangle$ , $\mathbf { v } = \langle - 6,5 \rangle$ to find the indicated quantity.State whether the result is a vector or a scalar. $( \mathbf { u } \cdot \mathbf { v } ) \mathbf { v }$

A force of F pounds is required to pull an object weighing W pounds up a ramp inclined at θ degrees from the horizontal. Find F if W = 100 pounds and θ = 11°.Approximate the answer to one decimal place.

To determine the distance between two aircraft, a tracking station continuously determines the distance to each aircraft and the angle A between them (see figure) .Determine the distance a between the planes when A = 44°, b = 37 miles, and c = 22 miles.

Use the vectors $\mathbf { u } = \langle 2,2 \rangle$ , $\mathbf { v } = \langle - 2,2 \rangle$ to find the indicated quantity.State whether the result is a vector or a scalar. $3 \mathbf { u } \cdot \mathbf { v }$