Trig Final Exam Review F07 O’Brien
Trigonometry, 8th ed; Lial, Hornsby, Schneider
Trigonometry Final Exam Review: Chapters 7, 8, 9
Note:
A portion of Exam 3 will cover Chapters 1 – 6, so be sure you rework problems from the first and second exams and
from the Exam 1 and Exam 2 Reviews.
Directions:
Work these problems with no resources other than the departmental formula sheet and a graphing calculator.
Your book, notebook, homework, solutions manual, etc. should be closed. Read and carefully follow all
directions. Show all of your work on all problems. Anytime you are asked to perform a calculation
manually or to give an exact answer, you may not use a calculator.
Chapter 7: Applications of Trigonometry and Vectors
1.
Determine the remaining sides and angles of the triangle ABC. Show all work and / or support your answer.
2.
Determine the remaining sides and angles of the triangle ABC. Show all work and / or support your answer.
C = 71.83°, B = 42.57°, a = 2.614
3.
A ship is sailing due north. At a certain point the bearing of a lighthouse 12.5 km distant is N 38.8° E. Later on, the
captain notices that the bearing of the lighthouse has become S 44.2° E. How far did the ship travel between the two
observations of the lighthouse? Show all work and / or support your answer.
4.
Mark notices that the bearing of a tree on the opposite bank of a river flowing north is 115.45°. Lisa is on the same
bank as Mark, but 428.3 m away. She notices that the bearing of the tree is 45.47°. The two banks are parallel. What
is the distance across the river? Show all work and / or support your answer.
5.
Find the area of the triangle ABC. Show all work and / or support your answer.
A = 30.50°, b = 13.00 cm, C = 112.60°
6.
A real estate agent wants to find the area of a triangular lot. A surveyor takes measurements and finds that two sides
are 52.1 m and 21.3 m, and the angle between them is 42.2°. What is the area of the triangular lot? Show all work and /
or support your answer.
7.
Determine the number of triangles ABC possible with the given parts. Show all work and / or support your answer.
a) b = 60, a = 82, B = 100°
8.
b) a = 35, b = 30, A = 40°
c) B = 54°, c = 28, b = 23
Find the unknown angles in triangle ABC, if the triangle exists. Show all work and / or support your answer.
a) C = 41°20', b = 25.9 m, c = 38.4 m
b) B = 48.2°, a = 890 cm, b = 697 cm
c) B = 74.3°, a = 859 m, b = 783 m
9.
Solve the triangle ABC, if possible. Show all work and / or support your answer.
C = 52.3°, a = 32.5 yd, c = 59.8 yd
1
Trig Final Exam Review F07 O’Brien
Trigonometry, 8th ed; Lial, Hornsby, Schneider
10.
Solve the triangle ABC, if possible. Show all work and / or support your answer.
A = 38°40', a = 9.72 km, b = 11.8 km
11.
Solve each triangle and then find the area of the triangle. Approximate values to the nearest tenth. Show all work
and / or support your answer.
a)
12.
b)
Solve each triangle and then find the area of the triangle. Show all work and / or support your answer.
a) C = 28.3°, b = 5.71 in., a = 4.21 in.
b) a = 189 yd, b = 214 yd, c = 325 yd
13.
The sides of a parallelogram are 4.0 cm and 6.0 cm. One angle is 58° while another is 122°. Find the lengths of the
diagonals of the parallelogram. Show all work and / or support your answer.
14.
A weight is supported by cables attached to both ends of a balance beam,
as shown in the figure. What angles are formed between the beam and the
cables? Show all work and / or support your answer.
15.
Using the vectors in the figure below, draw a sketch to represent:
a) c + d
b) d – c
c)
3c
16.
Use the vectors in the adjacent figure to find a) a + b, b) a – b, c) – a.
Show all work and / or support your answer.
17.
For vectors u and w with angle θ between them, sketch the resultant, u + w.
u  8 , w  12 , θ = 20°
18.
Find the magnitude and direction angle for vector v = 15, - 8 . Show all work and / or support your answer.
19.
Vector v has magnitude v  15.4 and direction angle α  27  30  . Find the horizontal (a) and vertical (b)
components of v. Show all work and / or support your answer.
2
Trigonometry, 8th ed; Lial, Hornsby, Schneider
Trig Final Exam Review F07 O’Brien
20.
Write vector v in component form, a, b .
Show all work and / or support your answer.
21.
Two forces act at a point in the plane. One is 19
newtons, the other is 32 newtons. The angle
between the forces is 118°. Find the magnitude of the
resultant force. Show all work and / or support your answer.
22.
Use the parallelogram rule to find the magnitude of the resultant force for
the two forces shown in the adjacent figure. Round your answer to the
nearest tenth. Show all work and / or support your answer.
23.
Given u =  2, 5 and v = 4, 3 , find –2u + 4v. Write your answer in the form ai + bj.
Show all work and / or support your answer.
24.
Find the dot product of 2, - 3 and 6, 5 . Show all work and / or support your answer.
25.
Find the angle between the vectors -5i + 12j and 3i + 2j. Show all work and / or support your answer.
26.
Given u =  2, 1 , v = 3, 4 , and w =  5, 12 , evaluate u  v  w  . Show all work and / or support your answer.
27.
Determine whether the vectors 1, 2 and  6, 3 are orthogonal. Show all work and / or support your answer.
28.
Two tugboats are pulling a disabled speedboat into port with forces 1240 lb and 1480 lb. The angle between these
forces is 28.2°. Find the magnitude and direction of the equilibrant. Show all work and / or support your answer.
29.
Two forces of 128 lb and 253 lb act at a point. The equilibrant force is 320 lb. Find the angle between the forces.
Show all work and / or support your answer.
30.
A force of 176 lb makes an angle of 78°50' with a second force. The resultant of the two forces makes an angle of
41°10' with the first force. Find the magnitudes of the second force and the resultant. Show all work and / or support
your answer.
31.
A 130-lb force keeps a 600-lb object from sliding down an inclined ramp. Find the angle of the ramp.
Show all work and / or support your answer.
32.
Find the magnitude of force necessary to keep a 3500-pound car from sliding down a ramp of 5°.
Show all work and / or support your answer.
33.
Two people are carrying a box. One person exerts a force of 150 lb at an angle of 62.4° with the horizontal. The
other person exerts a force of 114 lb at an angle of 54.9°. Find the weight of the box. Show all work and / or support
your answer.
3
Trig Final Exam Review F07 O’Brien
Trigonometry, 8th ed; Lial, Hornsby, Schneider
34.
A ship leaves port on a bearing of 34.0° and travels 10.4 miles. The ship then turns due east and travels 4.6 miles.
How far is the ship from port, and what is its bearing from port? Show all work and / or support your answer.
35.
A pilot is flying at 168 mph. She wants her flight path to be on a bearing of 57°40'. A wind is blowing from the south
at 27.1 mph. Find the bearing the pilot should fly, and find the plane’s groundspeed. Show all work and / or support
your answer.
36.
A pilot wants to fly on a bearing of 74.9°. By flying due east, he finds a 42-mph wind, blowing from the south, puts
him on course. Find the groundspeed and airspeed. Show all work and / or support your answer.
Chapter 8: Complex Numbers, Polar Equations, and Parametric Equations
 288 as the product of a real number and i. Show all work and / or support your answer.
37.
Write
38.
Solve 3x 2  2  4x and express all nonreal complex solutions in terms of i. Show all work and / or support your
answer.
39.
Perform the indicated operations and then simplify. Show all work and / or support your answer.
a)
40.
b)
 40
 12   6
8
Perform the indicated operation and then simplify. Write your answers in standard form. Show all work and / or
support your answer.
a)
41.
10
 4  i   2  3i    4  5i 
b)
 2  3i 4  2i 
c)
2  3i
1  5i
Simplify each power of i. Show all work and / or support your answer.
a)
b) i 13
i 25
42.
Find the sum of 5  6i and 2  3i by graphing each number in the complex plane and finding their resultant.
Show all work and / or support your answer.
43.
Write the complex number 3[cos 150° + i sin 150°] in rectangular form. Show all work and / or support your answer.
44.
Write the complex number –5 – 5i in trigonometric form r(cos θ + i sin θ), with θ in the interval [0°, 360°).
Show all work and / or support your answer.
45.
Find each product and write it in rectangular form. On part b, round to four decimal places. Show all work and / or
support your answer.
a)
8cos 300

  
 i sin 300   5 cos120   i sin 120 

b) (12 cis 18.5°) · (3 cis 12.5°)
4
Trig Final Exam Review F07 O’Brien
Trigonometry, 8th ed; Lial, Hornsby, Schneider

2cos150
4 cos120   i sin 120 
 & write it in rectangular form. Show all work and / or support your answer.

46.
Find the quotient
47.
Convert the numerator and denominator of
48.
Find each power. Write each answer in rectangular form. Show all work and / or support your answer.
53.
 3
b)
 2  2i 5
27 cis 300 
b)
3 i
x4 i  0
b) x 4  16  0
For each pair of polar points, plot the point; give two other pairs of polar coordinates (including one with a negative r);
and give the rectangular coordinates for the point. Show all work and / or support your answer.
a)
52.
3 cis 100 
Find all complex solutions of each equation. Leave answers in trigonometric form. Show all work and / or support
your answer.
a)
51.
2i
Find all cube roots of each complex number. Leave answers in trigonometric form. Show all work and / or support
your answer.
a)
50.
 i sin 150

to trigonometric form. Then find the quotient and write it in
1  i 3
rectangular form. Show all work and / or support your answer.
a)
49.

 2, 135 

5 

b)  3, 

3 


3
1
Given the rectangular coordinates  
,   , give two pairs of polar coordinates for the point, including one with a
 2
2 

negative r. Select θ so it is in the interval [0°, 360°). Show all work and / or support your answer.
For each rectangular equation, give its equivalent polar equation and sketch its graph. Show all work and / or support
your answer.
a) 3x – 2y = 6
b) x 2  y 2  9
5
Trig Final Exam Review F07 O’Brien
Trigonometry, 8th ed; Lial, Hornsby, Schneider
54.
Match each equation with its corresponding graph.
a.
r = a cos θ
A.
b.
r = a sin θ
B.
c.
r = a ± b sin θ with
a
1
b
C.
d.
r = a ± b cos θ with
a
1
b
D.
e.
r = a cos 4θ
E.
f.
r = a sin 5θ
F.
g.
r=
h.
r2 = 4 sin 2θ
c
acos θ  b sin θ
G.
H.
6
Trig Final Exam Review F07 O’Brien
Trigonometry, 8th ed; Lial, Hornsby, Schneider
55.
Identify the type of polar equation and draw a complete graph. Include a t-chart with at least 5 key points with your
graph. Show all work and / or support your answer.
a) r = 2 + 2 cos θ
56.
b) r = 4 cos 2θ
For each polar equation, give its equivalent rectangular equation and sketch its graph. Include a t-chart with at least
5 key points with your graph. Show all work and / or support your answer.
3
a) r = 2 sin θ
b) r 
1  sinθ
Chapter 9: Exponential and Logarithmic Functions
57.
Graph the following exponential function. Give the equation of the horizontal asymptote, a 5-point t-chart with the
“key point” at the center, and an accurate graph with one notch on each axis labeled. Show all of your work.
y  2 x 4   1
58.
Solve the following exponential equations algebraically. Show all your work and give an exact answer.
a.
2
 
3
59.
Find the balance A for P dollars invested at rate r for t years and compounded n times.
a.
P = $2000, r = 6.5%, t = 10 years, compounded quarterly.
b.
P = $3000, r = 7%, t = 8 years, compounded continuously.
60.
Find the required annual interest rate to the nearest tenth of a percent for $48,000 to grow to $78,186.94 when
compounded semiannually for 5 yr. Round your decimal answer to four decimal places and convert to a percent.
Show all of your work.
61.
Solve the following logarithmic equations algebraically. Show all your work and give an exact answer.
1
log x
 2
b. log x  5
c. log 4  x
3
16
16
a.
62.
x

9
4
b.
2x  4  8x  6
3
c.
x 4  125
Graph the following logarithmic function. Give the equation of the vertical asymptote, the exponential form of the
equation, a 5-point t-chart with the “key” point at the center, and an accurate graph with one notch on each axis
labeled. Show all of your work.
y = log 3 x  2  4
63.
Use the properties of logarithms to expand or condense the following log expressions. Show all of your work.
a.
log b
y3 x
z
b.
4
1
2
2 log a x  log a y  log a z
3
3
64.
Find the pH for lye soap which has a hydronium ion concentration of 3.2 10 14 moles per liter.
65.
Find the H 3 O  for drinking water which has a pH of 6.5.


7
Trig Final Exam Review F07 O’Brien
Trigonometry, 8th ed; Lial, Hornsby, Schneider
66.
Use the change of base theorem to find an approximation for log 1 15 . Round your answer to four decimal places.
2
67.
Given f(x) = 3 x, evaluate the following. Show all of your work.
a.
f (log 3 7)
68.
a.
Find the decibel rating of a sound having an intensity of 100,000 I0.
b.
If the intensity of the sound is doubled, by how much is the decibel rating increased?
a.
Find the Richter scale rating for an earthquake having an amplitude of 1,000,000 I 0.
b.
Express the magnitude of a 6.7 earthquake (on the Richter scale) in terms of I0.
69.
70.
71.
b.
f [log 3 (ln 3)]
c.
f [log 3 (2 ln 3)]
Solve.
a.
4 3x1  6
a.
c.
2 x 5  3 x
 .06 
1000 1 

4 

c.
6e 4x3  1  11
log (2x – 1) + log 10x = log 10
b.
ln x + ln (x – 1) = 2
log 6 4x  log 6 x  3  log 6 12
d.
log (11x + 9) = 3 + log (x + 3)
b.
d.
4t
 5000
72.
Elena McDuff wants to buy an $8,000 painting. She has saved $6,000. Find the number of years (to the nearest
tenth) it will take for her savings to grow to $8,000 at 5.7% compounded monthly.
73.
At what interest rate will $2,500 grow to $4,671.04 if invested for 10 years and interest is compounded quarterly?
Answers
Chapter 7: Applications of Trigonometry and Vectors
1.
A = 37.2°, a = 178 m, c = 244 m
2.
A = 65.60°, b = 1.942 cm, c = 2.727 cm
3.
27.8 km
4.
293.4 m
5.
65.94 cm2
6.
373 m2
7.
a) 0;
8.
a) B ≈ 26°30', A ≈ 112°10';
9.
A ≈ 25.5°, B ≈ 102.2°, b ≈ 73.9 yd
10.
B1 ≈ 49°20', C1 ≈ 92°00', c1 ≈ 15.5 km, B2 ≈ 130°40', C2 ≈ 10°40', c2 ≈ 2.88 km
b) 1;
c) 2
b) A1 ≈ 72.2°, C1 ≈ 59.6°, A2 ≈ 107.8°, C2 ≈ 24.0°;
c) no such triangle
8
Trig Final Exam Review F07 O’Brien
Trigonometry, 8th ed; Lial, Hornsby, Schneider
11.
a) A ≈ 22.3°, B ≈ 108.2°, C ≈ 49.5°; area =
b) A ≈ 56.7°, C ≈ 68.3°, b ≈ 88.2; area =
12.
231  15.2 units 2
13547079  3680.6 units 2
a) c ≈ 2.83 in., A ≈ 44.9°, B ≈ 106.8°; area =
b) C ≈ 107°20', B ≈ 39°00', A ≈ 33°40'; area =
13.
5.2 cm, 8.8 cm
14.
26.4° and 36.3°
15.
a)
16.
a)
32.53688517  5.7 units 2
372645000 19304.0 units 2
b)
0, 10
b)
c)
8, 2
c)
 4, - 6
17.
18.
v  17 ; θ = 331.9°
19.
a ≈ 13.7, b ≈ 7.11
20.
v = 4, 4 3
21.
29 newtons
22.
24.4 lb
23.
20i + 2j
24.
–3
25.
78.93°
26.
–24
27.
orthogonal
28.
2640 lb at an angle of 167.2° with the 1480-lb force
29.
70.1°
30.
magnitude of second force: 190 lb; magnitude of the resultant: 283 lb
31.
12.5°
32.
305 lbs
33.
226 lb
34.
13.5 mi; 50.4°
35.
bearing: 65°30'; groundspeed: 181 mph
36.
groundspeed: 161 mph; airspeed: 156 mph
9
Trig Final Exam Review F07 O’Brien
Trigonometry, 8th ed; Lial, Hornsby, Schneider
Chapter 8: Complex Numbers, Polar Equations, and Parametric Equations
37.
12 2 i
38.
x
39.
a)
1
i
2
b) –3
40.
a)
10  i
b)
41.
a) i
43.

44.
5 2 cos225   i sin225 
45.
a)
2
2

i
3 3
2  16i
b) – i
3 3 3
 i
2
2

42.
3  3i ;

20  20i 3
b) 30.8580 + 18.5414 i
3 1
 i
2 2
47.

48.
a)
49.
a) 3 cis 100°, 3 cis 220°, 3 cis 340°
3
27 27 3

i
2
2

2 cos 350   i sin 350 
b) 128 + 128i




b) 3 2 cos 110   i sin 110  , 3 2 cos 230   i sin 230  ,

a) cos 22.5° + i sin 22.5°, cos 112.5° + i sin 112.5°, cos 202.5° + i sin 202.5°, cos 292.5° + i sin 292.5°
b) 2 cis 45°,
51.
1 1
 i
2 2
3 i
46.
50.
c) 
a)
b)
2 cis 135°, 2 cis 225°, 2 cis 315°
 2, 495 , (2, 315°),  2 ,  2 

4π   3 3 3 
 π 
,
 3,  ,   3,
,
3   2 2 
 3 
52.
(1, 210°), (-1, 30°)
53.
a)
r
54.
a.
F
6
3cosθ  2sinθ
b.
A
b) r = 3 or r = –3
c.
D
d.
G
e.
H
f.
B
g. C
h.
E
10
Trig Final Exam Review F07 O’Brien
Trigonometry, 8th ed; Lial, Hornsby, Schneider
55a.
55b.
56a.
56b.
57.
HA: y = 1
“key point”: (4, 0)
x
2
3
4
5
6
y
3
4
1
2
0
-1
-3
58.
a.
x=2
b.
x = 11
59.
a.
$3811.12
b.
$5252.02
60.
10%
61.
a.
x=4
b.
x = 243
c.
x = 625
c.
x
1
2
11
Trig Final Exam Review F07 O’Brien
Trigonometry, 8th ed; Lial, Hornsby, Schneider
62.
VA: x = –2; exponential form of the equation: x  3 y  4  2
x
 17
9
5
3
-1
1
7
63.
a.
3 log b y 
y
-6
-5
-4
-3
-2
1
log b x  4 log b z
2
b.
log a
x2
3
64.
13.5
65.
3.2  10 7
66.
–3.9069
67.
a.
7
b.
ln 3  1.0986
68.
a.
50
b.
about 3 decibels
69.
a.
6
b.
about 5,000,000 I0
70.
a.
.7642
b.
71.
a.
1
b.
72.
5.1 years
73.
6.3%
yz 2
c.
2 ln 3 = ln 32 = ln 9  2.1972
8.5476
c.
–.5767
d.
27.0246
3.2639
c.
4.5
d.
Ø
12