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° 
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 
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)  $b \geq 304.6 , b \geq \frac { 304.6 } { \sin 62 ^ { \circ } }$ 
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 
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)     
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 
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)   $\mathbf { v } = \left\langle \frac { 54 \sqrt { 61 } } { 61 } , - \frac { 45 \sqrt { 61 } } { 61 } \right\rangle$ 
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° 
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 
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 
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)  $\langle 0,1 \rangle$ 
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)  $38.43 ^ { \circ }$ 
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 
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 
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 
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)   $Z = 3 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$ 
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 
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 
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 
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 
A)  $\langle 0,2 \rangle$ ; vector 
B) -2; scalar 
C) 2; scalar 
D)  $\langle 4 , - 4 \rangle$ ; vector 
E) 0; scalar

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

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. =\langle6,2\rangle =\langle7,0\rangle (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.

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

Functions and Their Graphs

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Trigonometry

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Topics In Analytic Geometry

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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. =\langle6,2\rangle =\langle7,0\rangle (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

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometry

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

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. =\langle6,2\rangle =\langle7,0\rangle (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. $3 \mathbf { u } \cdot \mathbf { v }$