APPLICATIONS
OF TRIGONOMETRY
Algebra 2 & Trigonometry Lab
Miss Kersting
Room 206
Period 1
Name:____________________________
Day 1 – Law of Cosines
To find the missing side of a triangle if the other two sides and the angle between the
two sides are known, use Law of Cosines. (SAS)
a 2  b 2  c 2  2bc cos A
1.)
*Note: “a” is the unknown side
Given triangle ABC, where angle B is 60 degrees, sides AB = 6 and BC = 4. Find the third side
of the triangle in simplest radical form.
2.)
Given triangle ABC, where angle A is 80 degrees, sides AB = 5 and AC = 13. Find the third
side of the triangle to the nearest tenth.
3.)
Given triangle PQR, p = 4, r = 6 and cos Q 
1
, find q.
16
p2
To find an angle of a triangle if all three sides are known, use Law of Cosines. (SAS)
a 2  b 2  c 2  2bc cos A
*Note: “a” is the side opposite the unknown angle
4.)
In ABC a = 5, b = 9, and c = 12, find cos A.
5.)
In RED , RE = ED = 5, RD = 8. Find the measure of the largest angle to the nearest degree.
6.)
Stephen is working on a stained-glass project and needs to form a triangle with sides 8, 12,
and 15 inches out of lead cane to enclose the glass. To the nearest tenth of a degree, what is
the largest angle he needs to create using the lead caning?
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Directions: Answer each question.
1.) In triangle ABC, a = 5, b = 7, and c = 8.
Find cos C .
Day 1 – Law of Cosines
HOMEWORK
2.)
In triangle RST, r = 5, s = 7 and cos T 
1
.
2
Find t.
3.)
In triangle ABC, the measures of the sides
are a  3 , b  5 , and c  7 . Find the
measure of the largest angle in the
triangle.
4.)
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Find, to the nearest tenth of a centimeter,
the measure of the base of an isosceles
triangle whose equal sides each measure
3cm and whose vertex angle measures 50°.
5.)
A cross-country ski trail is laid out in the shape of a triangle. The lengths of the three paths
that make up the trail are 2000 meters, 1200 meters, and 1800 meters. Find, to the nearest
degree, the measure of the smallest angle of the trail.
Answers to Day 1 Homework:
1
1.)
7
2.) 39
3.) 120
4.) 2.5cm
5.) 36
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Day 2 – Law of Sines and area of triangles/parallelograms
Do Now: Question 1
1.) Al is standing 50 yards from a maple tree and 30 yards from an oak tree in the park. His
position is shown in the accompanying diagram. If he is looking at the maple tree, he needs to
turn his head 120° to look at the oak tree. How many yards apart are the two trees?
To find a side of a triangle if one side and two angles are known, use Law of Sines.
(AAS, ASA)
a
b
c


sin A sin B sin C
2.) In ABC , b = 12, sin B = 0.6, and sin C = 0.9. Find c.
𝜋
𝜋
3.) In ABC , if a = 9, 𝑚 < 𝐴 = 3 , and 𝑚 < 𝐵 = 4 , find the exact value of b in simplest form.
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Steps to find the Area of a Triangle
1.) You must have SAS (the included angle). If you do not have side SAS use the Law of Sines or
Cosines to find a side or angle.
1
2.) Plug in the following formula to obtain the area: K  ab sin C .
2
1
4.) In ABC , a = 16, b = 12, and sin C  . Find the area of ABC.
2
5.) Find the area of the triangle to the nearest hundredth of a square meter.
6.) In parallelogram HOPE, HO = 16.7 centimeters, HE = 20.1 centimeters, and mOHE  108.3 .
a.) Find the length of OE to the nearest tenth of a centimeter.
b.) Find the area of parallelogram HOPE to the nearest hundredth of a square
centimeter.
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Day 2 – Law of Sines and area of triangles/parallelograms
HOMEWORK
𝜋

DEF
2.)
In
,
sin
D
=
0.4,
sin
E
=
0.25
1.) In ABC , if a = 24, 𝑚 < 𝐴 = 6 ,
and d = 20. Find the exact value of e.
𝜋
and 𝑚 < 𝐵 = 2 , find the exact value of b in
simplest form.
3.) The length of one of the equal sides of an isosceles triangle measures 25.8 inches and each base
angle measures 53 degrees.
a.) Find the measure of the base of the triangle to the nearest tenth.
b.) Find the perimeter of the triangle to the nearest inch.
4.) A ski lift begins at ground level 0.75 mile from the base of a mountain whose face has a 50°
angle of elevation, as shown in the accompanying diagram. The ski lift ascends in a straight
line at an angle of 20°. Find the length of the ski lift from the beginning of the ski lift to the top
of the mountain, to the nearest hundredth of a mile.
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5.) Each base angle of an isosceles triangle measures 57.4  and the equal sides measure 10.8
inches. Find the area of the triangle to the nearest hundredth of a square inch.
6.) If the area of an isosceles triangle is 68 square feet and the vertex angle is 66  , what of the
length of an equal side of the triangle?
7.) In parallelogram JACK, JCK measures 6115' , JC = 23.5, and KC = 18.7. What is the area of
the parallelogram to the nearest tenth?
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Day 3 – Ambiguous Case
Do Now: Question 1
1.) The accompanying diagram shows the approximate linear distances traveled by a sailboat
during a race. The sailboat started at point S, traveled to points E and A, respectively, and
ended at point S. Based on the measures shown in the diagram, which equation can be used
to find x, the distance from point A to point S?
(1)
x
sin 75

sin 65
32
(2)
sin 65 sin 75

x
32
(3)
x 32

65 75
(4)
65 32

x 75
Steps to Finding the Number of Distinct Triangles
1.) The number of possible triangles will be either 0, 1, or 2.
2.) Set up an equation using the Law of Sines to find the missing angle measure.
3.) Determine the two possible degree measures for the angle, since the angle could be in
quadrant I or II. (Sine is positive in quadrants I and II.)
2.) How many distinct triangles can you form, if a = 5, b = 6, and the mA  30  ?
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3.) How many distinct triangles can you form, if a = 4, b = 6, and the mB  150  ?
4.) How many distinct triangles can you form, if a = 2 , b = 3, and the mA  45  ?
5.) Multiple Choice:
In PQR , mP  30  , p = 5, r = 8, then PQR is
(1) an acute triangle
(2) a right triangle
(3) an obtuse triangle
(4) an acute or an obtuse triangle
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Day 3 – Ambiguous Case
HOMEWORK
Directions: For questions 1 – 4, determine how many non-congruent triangles can be constructed.
1.) In ABC, a = 4, b = 6, and mA = 30º.
2.) In ABC, a  2 , b = 3, and mA = 45º.
3.) In ABC, a = 6, b = 8, and mA = 30º.
4.) In ABC, a = 5, b = 10, and mA = 30º.
5.) The measures of two sides of a triangle are 34 and 22, and the measure of the angle opposite
the smaller of these sides is 30º. Find to the nearest degree two possible measures of the
angle opposite the larger of these sides.
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Day 4 – Solving triangles, including double triangles and word problems
Do Now: Question 1
1
1.) In  ABC, AC = 18, BC = 10, and cos C  . Find the area of  ABC to the nearest tenth of a
2
square unit.
Steps to Solve Double Triangle Problems
1.
2.
3.
4.
2.)
Draw the picture, if it’s not given.
Find the measure of all angles.
Find the length of the middle line using either the Law of Sines or Law of Cosines.
Find the height of the wanted line using the new middle line and Law of Sines.
The accompanying diagram shows the plans for a cell-phone tower that is to be built near a
busy highway. Find the height of the tower, to the nearest foot.
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3.)
While sailing a boat offshore, Donna sees a lighthouse and calculates that the angle of
elevation to the top of the lighthouse is 3°. When she sails her boat 700 feet closer to the
lighthouse, she finds that the angle of elevation is now 5°. How tall, to the nearest tenth of a
foot, is the lighthouse?
Steps to Solve Double Triangle Problems
1.
2.
4.)
Draw a diagram of a parallelogram. The two applied forces are the adjacent sides of the
parallelogram, and the diagonal is called the resultant.
Solve using Law of Sines or Law of Cosines, or if needed, like a Double-Triangle problem.
Two forces of 25 and 15 pounds act on a body so that the angle between them is 75°. Find, to
the nearest pound, the magnitude of the resultant.
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5.)
A resultant force of 143 pounds is needed to move a heavy box. Two applied forces act at
angles of 3540' and 4730' with the resultant. Find, to the nearest pound, the magnitude of
the larger force.
6.)
Two forces of 40 pounds and 20 pounds, respectively, act simultaneously on an object. The
angle between the two forces is 40°.
a.) Find the magnitude of the resultant, to the nearest tenth of a pound.
b.) Find the measure of the angle, to the nearest degree, between the resultant and the
larger force.
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1.)
2.)
Day 4 – Solving triangles, including double triangles
HOMEWORK
Two forces of 28 pounds and 41 pounds act on a body so the angle between the two forces
measure 72  . Find, to the nearest tenth of a pound, the magnitude of the resultant the forces
produce.
Two applied forces produce a resultant force of 18.6 pounds. The smaller force measures 15.8
pounds, and the larger force is 24.3 pounds. Find the measure of the angle between the two
forces to the nearest tenth of a degree.
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3.)
A resultant of 162 pounds must be exerted to move a refrigerator. If two applied forces act on
the refrigerator at angles of 43.6  and 38.7  , respectively, find the magnitude of each of the
two applied forces, to the nearest tenth of a pound.
4.)
A ship at sea heads directly toward a cliff on the shoreline. The accompanying diagram shows the top
of the cliff, D, sighted from two locations, A and B, which are separated by distance S. If
,
, and
, what is the height of the cliff, to the nearest foot?
Day 4 Homework Solutions:
1.) 56.3 lbs; 2.) 49.9°;
3.) 102.2 lbs and 112.7 lbs;
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4.) 41 feet
1.)
Day 5 – Combination Day
Firefighters dug three trenches in the shape of a triangle to prevent a fire from completely
destroying a forest. The lengths of the trenches were 250 feet, 312 feet, and 490 feet.
Find, to the nearest degree, the smallest angle formed by the trenches.
Find the area of the plot of land within the trenches, to the nearest square foot.
2.)
A tractor stuck in the mud is being pulled out by two trucks. One truck applies a force of
1,200 pounds, and the other applies a force of 1,700 pounds. The angle between the forces
applied by the two trucks is 72°. Find the magnitude of the resultant force, to the nearest
pound.
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3.)
A jet is flying at a speed of 526 miles per hour. The pilot encounters turbulence due to a 50miles-per-hour wind blowing at an angle of 47  , as shown in the accompanying diagram.
Find the resultant speed of the jet, to the nearest tenth of a mile per hour.
Use this answer to find the measure of the angle between the resultant force and the wind
vector, to the nearest tenth of a degree.
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4.) As shown in the accompanying diagram, two tracking stations, A and B, are on an east-west line
110 miles apart. A forest fire is located at F, on a bearing 42 northeast of station A and 15
northeast of station B. How far, to the nearest mile, is the fire form station A?
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