Pre Calculus
Worksheet 5.5
Solve each triangle using the Law of Sines.
1. ∆ABC where B = 16°, C = 103°, c = 12
2. ∆WXY where X =81°, Y =59°, w =92
3. How can you tell, without using the Law of Sines, that a triangle cannot be formed by using the measurements
A = 89°, B =104°, a = 28 ? Explain.
4. Use ∆PQR at the right to answer the questions below.
Q
10cm
a) Draw the altitude to PR and label it QT.
b) Write an expression for the length of QT. Do not find
the actual length.
P
R
17cm
c) Find the area of ∆PQR .
5. To find the length of the span of a proposed ski lift from A to B, a surveyor measures the angle DAB to be 25° and then
walks off a distance of 1000 feet to C and measures the angle ACB to be 15°. What is the distance from A to B ?
B
25°
D
A
1000 ft
C
6. An emergency dispatcher must determine the position of a caller reporting a fire. Based on the caller’s cell phone
records, she is located in the area shown. Overcome by the desire to solve for any missing lengths, the dispatcher
momentarily forgets about the fire and wants to know what the unknown side lengths are in the triangle.
Tower
2
60.1°
52.4°
67.5°
Tower
1
4.6 mi
Tower
3
7. An adventurer who is stuck on the top of a cliff is trying to decide whether or not he can jump to the next ledge. In his
free time he is able to determine the angle from the bottom of the tree to the edge of the cliff and the angle from the top of
the tree to the edge of the cliff as shown in the diagram below. While he was climbing the tree to measure that second
angle he figured out that the tree is 13 feet tall. Find the distance d the adventurer will have to jump in order to make it to
the ledge (assuming he doesn’t trip over the roots of the tree and fall to the bottom of the cliff).
38°
125°
d
8. A buoy is anchored offshore to mark a sandbar. The straight shoreline at that location runs north and south. From two
observation points on the shore 2.4 miles apart, the bearings to the buoy are 134° and 22°.
a) What is the distance from the buoy to each of the observation points?
b) How far is the buoy from the shore?
9. US 41, a highway whose primary directions are north-south, is being constructed along the west coast of Florida. Near
Naples, a bay obstructs the straight path of the road. Since the cost of a bridge is prohibitive, engineers decide to go
around the bay. The illustration shows the path that they decide on and the measurements taken. What is the length of
highway needed to go around the bay?
140°
Pelican
2 mi
Bay
135°
US
41
1/8 mi
1/8 mi
Pre Calculus
Worksheet 5.6
Solve each triangle using the Law of Cosines.
1. ∆ABC where B = 131°, a = 13, c = 8
2. ∆WXY where
=
x 28,
=
y 17,
=
w 30
Determine whether to use SohCahToa, Law of Sines or Law of Cosines to solve triangle DEF. Explain.

3.
=
f 15,
=
D 31=
, E 42
4.=
d 32,
=
e 42,
=
f 13
5. D = 28 , E = 98°, d = 6
6. =
f 3,=
e 5.5,=
D 40
7. d= 11, D= 22 , E= 68°
8.=
f 27,
=
e 20,
=
E 119
9. Give two reasons why you cannot have a triangle with sides 13 cm, 9 cm and 4 cm.
10. Find the area of the triangle in question 6.
11. Find the area of the triangle in question 4.
12. Two planes that were flying together in formation take off in different directions. One plane goes East at 350 mph,
and the other plane goes ENE at 380 mph. (The angle between E and ENE is 22.5° ...  “you’re welcome”). How far
apart are the planes two hours after they separate?
13. US 41, a highway whose primary directions are north-south, is being constructed along the west coast of Florida.
Near Naples, a bay obstructs the straight path of the road. Since the cost of a bridge is prohibitive, engineers decide to go
around the bay. The illustration shows the path that they decide on and the measurements taken. What is the length of
highway needed to go around the bay?
140°
Pelican
2 mi
Bay
1/8 mi
1/8 mi
135°
US
41
14. An airplane flies north from Ft. Myers to Sarasota a distance of 150 miles, and then changes his bearing to 50° and
flies to Orlando, a distance of 100 miles.
a) How far is it from Ft. Myers to Orlando?
b) What bearing is needed for the pilot to return from Orlando to Ft. Myers?
15. Solve the following equation for P: p 2  h 2  d 2  2hd cos  P 
16. A streetlight is designed as shown below. Determine the angle in the design.
3
θ
2
4.5
Pre Calculus
Worksheet 5.5 (ambiguous case)
1. Explain why given SSA is called the ambiguous case.
State whether the given measurements determine zero, one or two triangles ABC.
2. C = 120°, a = 18, c = 9
3. C = 36°, a = 17, c = 16
4. B = 82°, b = 17, c = 15
5. For any questions 2–4 that have TWO triangles, solve BOTH triangles.
Solve the triangle WXY with the given parts below. IF there are two triangles, SOLVE BOTH!!
6. Y = 103°, w = 46, y = 61
7. X = 57°, w = 11, x = 10
8. On Spring Break, Bob and his friends decide to go 4-wheeling off road in his new Jeep. The Jeep has a winch (a lifting
device with a cable) that is used to pull the Jeep in case it gets stuck. After driving too fast over a hill, Bob finds himself
stuck in the middle of a shallow stream. While wading through the stream to attach the cable to a tree a hill on the other
side of the stream (see diagram), Bob ponders the mathematics of his situation…He wonders what the angle of elevation
of the cable was before his Jeep was pulled to the edge of the stream.
100 ft
65 ft
37°
Pre Calculus
Review 5.5 to 5.6
Name: __________________
Block: ______
Calculator Allowed-SHOW ALL WORK FOR CREDIT
Find the area of each triangle.
1. a = 5 cm, b = 12cm, c = 13cm
2. c = 3.58m, b = 6.8m, A = 39°
Solve each triangle. If there is no solution, explain why. If there are two triangles, solve BOTH!
3. b = 40, c = 45, A = 51°
4. c = 125, b = 150, C = 25°
5. w = 7.5, x = 10, y = 12
6. w = 8, W = 79°, X = 33°
7. Given an example where three parts of a triangle are given, but no triangle can be formed. Explain your
answer including a diagram.
For questions 8-9, do NOT copy a question on any of the worksheets for 5.5 or 5.6!! Make your own problem!!
you may choose whether or not a diagram is given in the problem.
8. Create a new application question that whose solution is found using the Law of Sines. Then, solve your
problem.
9. Create a new application question that whose solution is found using the Law of Cosines. Then, solve your
problem.
2