Honors Pre-Calculus
Law of Sines & Cosines Applications
Name
Where appropriate, give angle measures to the nearest tenth of a degree and lengths of sides in simplest radical form
or to the nearest hundredth.
1.
Find the area of
if
Also find .
1.
Area=______________
________________
2.
3.
Find the measure of the largest angle in a triangle with sides having lengths
√
√ and √ .
2.
___________________
In
3.
________________
,
Find
.
________________
4.
Three measurements in
are given as
Show that at least one of the measurements is incorrect.
5.
A regular polygon with
sides is inscribed in a circle with radius .
Find its area. Compare your answer with .
6.
7.
5.
___________________
A submarine dives at an angle of
with the horizontal. If it takes minutes
to dive from the surface to a depth of
feet, how fast does it move along
its sloping path downward? Give your answer in feet per minute. Then convert
it to nautical miles per hour. Note: nautical mile per hour
feet per hour).
6.
___________________
In
,
Find the measure(s) of
.
7.
______________
8.
In parallelogram
9.
,
.
a.
Find the area of
.
b.
Find the lengths of both diagonals.
A triangle has an area of
and two of its sides are
and
Find the possible measures of the angle formed by these sides.
10.
In the diagram given below,
is similar to
a.
Find the lengths
b.
Find the ratio of the areas of the triangles.
C
a
and
8a.
___________________
8b.
___________________
9.
___________________
long.
.
10.
.
_______________
________________
________________
ratio = _____________
F
6
e
A
11.
B
D
f
E
The diagonals of a parallelogram have lengths and
and they meet
at a
angle. Find the area and perimeter of the parallelogram.
11.
Area =______________
Perimeter=__________
12.
An obtuse triangle with area
has two sides of lengths and
Find the length of the third side. There are two answers.
12.
___________________
13.
The perimeter of a regular decagon (
. Find its area.
13.
___________________
14.
If fencing costs
per foot, how much will it cost to buy fencing
to go around the plot of land shown below?
14.
___________________
ft
ft
sides) is
.
15.
16.
17.
In the township of Madison, rural undeveloped land is taxed at a rate
of
per acre. Find the tax on the plot of land shown.
Note: acre
ft2
15.
___________________
A ship is steaming north at knots ( nautical miles per hour) when the
captain sights a small island at an angle of
to the east of the ship’s
course. After
minutes, the angle is
. How far away is the island at
this moment (in nm)?
16.
___________________
In
,
a.
Solve
.
.
17a.
______________
______________
______________
18.
b.
Find the area of
.
c.
Find the length of the altitude to ̅̅̅̅ .
Ship sights ship on a compass bearing of
compass bearing of ship from ship .
19.
Ship
sights ship
on a bearing of
20.
An airplane flies on a course of
at a speed of
east of its starting point is it after 2 hours?
17b.
Area = _____________
17c.
___________________
18.
___________________
19.
___________________
20.
___________________
. Make a sketch and give the
. What is the bearing of
from ?
km/h. How far
21.
22.
A hunter walks east for hour and then north for
hours. What course
should the hunter take to return to his starting point? What assumptions
do you make to answer the question?
21.
___________________
Point is
km north of point , and point is
km north of point
a bearing of
from . Find the bearing and distance of from .
22.
Bearing_____________
on
Distance____________
23.
Point is km west of point , and point
Find the bearing and distance of from
is km southwest of .
23.
Bearing_____________
Distance____________
24.
25.
26.
Traveling at a speed of
knots, a ship proceeds south from its port
for
hours and then changes course to
for hours. At this time,
how far from port is the ship?
24.
___________________
A sailboat leaves its dock and proceeds east for miles. It then changes
course to
until it is due south of its dock. How far south is this?
25.
___________________
Two ships,
and , leave port at the same time. Ship proceeds at
knots
on a course of
, while ship proceeds at knots on a course of
.
After hours, ship loses power and radios for help. How far and on what
course must ship travel to reach ship .
26.
Bearing_____________
Distance____________
Sketch each plot of land described, and find its AREA.
27.
28.
29.
From an iron post, proceed
meters northeast to the brook, then
meters east along the brook to the old mill, then
meters
to a post on the edge of Wiggin’s Road, and finally along Wiggin’s Road
back to the iron post.
27.
Area=______________
From a granite post, proceed
ft east along Tasker Hill Road, then along a bearing
of
for
ft, then along a bearing of
for
ft, and finally along
a line back to the granite post.
28.
Area=______________
From a cement marker, proceed
m southwest to the river, then
m south
along the river to the bridge, then
m
to a sign on the edge of
Sycamore Lane, and finally along Sycamore Lane back to the cement marker.
29.
Area=______________
Answers: Law of Sines and Cosines Applications
1. Area =
;
2.
4. Sine cannot be greater than
5.
7.
8a. A =
9.
10a.
11. A =
sq. units
P=
units
14.
16.
nm
29.
6.
nmh
8b.
and
13.
sq. units
sq. ft.; Tax =
17a.
sq. units
or
23. Bearing:
,
10b.
12.
17c.
√
19.
21. Assumption: Rate is constant; Bearing:
26.
sq. units
15. A =
17b.
18.
sq. units
3.
; TR =
or
or
24.
27.
20.
km
22.
nm
25.
28.
km
miles
HPC
1.
Law of Sines and Cosines
Review for Quiz 6.1-6.2/Packet
Name______________________
An airplane flies on a course of 130 at a speed of 1100 km/h. How far east of its starting
point is it after 3 hours?
1.)___________________
2.
One angle of an isosceles triangle has a measure of 150. If the area of the triangle is
9 cm2, what is the perimeter of the triangle?
2.)___________________
3.
A ship leaves port and proceeds west 30 miles. It then changes course to 020 until it is
due north of its origin. How far north of its origin is it?
3.)___________________
4.
The area of ∆ABC is 45 square units. If a = 10 and b = 15, find the measure of angle C to
the nearest degree.
4.)___________________
5.
Given the diagram below, find ZN to the nearest whole unit.
5.)___________________
Y
2
N
5
4
Z
6.
7
X
Solve XYZ if x = 52, y = 70 and z = 100.
6.)___________________
_____________________
_____________________
7.
Solve XYZ and find its area if  X  52,  Y  70 , and z  100 .
7.)___________________
_____________________
_____________________
8.
Solve XYZ and find its area if  X  52, y  70, , and x  100 .
8.)___________________
_____________________
_____________________
9.
10.
11.
Eli Cooley flew his plane 900 km north, turned 15 , and flew
1150 km. How far is Mr. Cooley from his starting point?
9.)___________________
The angle of elevation to the peak of a mountain is 30 .
A kilometer closer, the angle of elevation is 35 . Find the
height of the mountain.
10.)__________________
A lakefront plot of land is shown below. What is its area and
lakefront footage (bolded sides are lakefront footage)?
11.)__________________
70
80
250 ft
350 ft
75
12.
The diagram shows the dimensions for a sail on a wooden model
ship. Find the area of the sail to the nearest square inch.
[The angle at the bottom left is NOT a right angle!]
12.)__________________
12.5 in
26 in
18 in.
75
16.5 in
13.
14.
15.
A ship leaves port and travels 36 miles west, then 24 miles on a
course bearing 213 . How far is it from its starting point?
13.)__________________
A hiker walks 8000 m on a course of S 81 E . She then changes
direction and hikes 5000 m on a course of N 32W . How far is she
from her starting point, and on what course must she travel to return
to the starting point?
14.)__________________
A regular pentagon is inscribed in a circle of radius 4 in. Find the area of the pentagon.
15.)__________________
16.
Observers at points A and B, 30 km apart, sight an airplane between them at angles of
elevation of 40 and 75, respectively. How far is the plane from each observer?
16.)__________________
17.
Two hikers follow a trail that splits into two forks. Each hiker takes a different fork. The forks
diverge at an angle of 67 and both hikers walk at a speed of 3.5 mph. How far apart are the
hikers after 1 hour?
17.)__________________
18.
After leaving an airport, a plane flies for 1.5 hours at a speed of 200 km/h on a course of
200. Then, on a course of 340, the plane flies for 2 hours at a speed of 250 km/h. At this
time, how far from the airport is the plane?
18.)__________________
6.  X  29.4,  Y  41.4,  Z  109.2
7. x  92.9, y  110.8,  Z  58, AREA  4364.9
8.  Y  33.5,  Z  94.5, z  126.5, AREA  3489.2 9. 2032.7 km
2
2
10. 3.3 km
11. Area = 55,132.21 ft , lakefront footage = 401.3 ft
12. 301.8 in
13. 53.04 miles
2
14. 6042.8 m, bearing of 240 or
15. 38.04 in
16. 31.97 km from A and 21.3 km from B 17. 3.86 miles
18. 331.9 km
1. 2527.9 km
2. 23.6 cm
3. 82.4 miles
4. 36.9 or 143.1
5. 5.0