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Triangles in Applied Problems
CK-12
Kaitlyn Spong
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Printed: May 11, 2016
AUTHORS
CK-12
Kaitlyn Spong
www.ck12.org
C HAPTER
Chapter 1. Triangles in Applied Problems
1
Triangles in Applied
Problems
Here you will apply your knowledge of trigonometry to solve problems related to triangles.
A community garden is being built in your neighborhood. The garden will be triangular in shape and a fence will
surround the garden. Two sides of the garden will be 33 feet and 24 feet. The angle between those two sides will
measure 62◦ . Find the area and perimeter of the garden.
Triangle Summary
There are many different problems you can solve with your knowledge of triangles and trigonometry. Here is a
summary of all the key facts and formulas that will be helpful.
TRIANGLE SUMMARY:
•
•
•
•
•
◦
The sum of the measures of the three angles in a triangle
√ is 180
:2
In 30-60-90 right triangles the sides are in the ratio 1 : 3√
In 45-45-90 right triangles the sides are in the ratio 1 : 1 : 2
The Pythagorean Theorem states that for a right triangle with legs a and b and hypotenuse c, a2 + b2 = c2
SOH CAH TOA is a mnemonic device to help you remember the three trigonometric ratios:
opposite leg
sin θ = hypotenuse
adjacent leg
cos θ = hypotenuse
opposite leg
tan θ = adjacent leg
• The Law of Sines: sina A = sinb B = sinC
c (watch out for the SSA case)
• The Law of Cosines: c2 = a2 + b2 − 2ab cosC
• The area of a triangle is 12 bh or 21 ab sinC
MEDIA
Click image to the left or use the URL below.
URL: https://www.ck12.org/flx/render/embeddedobject/58154
Solve the following problems
If a boat travels 4 miles SW (southwest) and then 3 miles NNW (north-northwest), how far away is the boat from its
starting point?
First, think about what southwest and north-northwest mean. The picture below shows the four basic directions
of north, south, east, and west. It also shows southwest and northwest. North-northwest is directly in between
northwest and north. Make sure you understand where the angles in the picture came from.
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Next, draw a picture of the situation.
Note that the angle between the SW direction and the NNW direction is 22.5◦ + 45◦ = 67.5◦ .
Now make a plan for how you will solve. Look to see what is given and what you are looking for and think about
what method or technique would be helpful. In this situation, you have two sides and an included angle and are
looking for the third side. You can use the Law of Cosines.
Now that you have a plan, you can solve the problem. For the Law of Cosines, the 3 and 4 are the values for a and
b and 67.5◦ is the value for 6 C. x is the side across from the angle, so it is c.
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Chapter 1. Triangles in Applied Problems
a2 + b2 − 2ab cosC = c2
32 + 42 − 2(3)(4)(cos 67.5) = x2
9 + 16 − 9.18 = x2
15.82 = x2
x ≈ 3.98
The boat is 3.98 miles from its starting place.
From the fourth story of a building (65 feet) Mark observes a car moving towards the building driving on the street
below. If the angle of depression of the car changes from 15◦ to 45◦ while he watches, how far did the car travel?
The angle of depression is the angle at which you view an object below the horizon. Start by making a detailed
picture of the situation and labeling what you know.
Note that 15◦ + 30◦ = 45◦ , the angle of depression for the ending location of the car.
Now make a plan for how you will solve. Look to see what is given and what you are looking for and think about
what method or technique would be helpful. In this situation, you have two right triangles (the smaller brown triangle
and the larger blue triangle). In each case, you know a side and an angle. You are looking for a portion of one of the
sides of these triangles (x).
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You can use trigonometric ratios to find the missing sides of these triangles to help you to find the length of x.
First look at the small brown triangle. y is opposite the 45◦ angle and 65 is adjacent to the 45◦ angle. This is a tangent
relationship (or you could use 45-45-90 triangle ratios):
y
65
y = 65 tan 45◦
tan 45◦ =
y = 65 f t
Now look at the larger right triangle. x + y is opposite the 75◦ angle (30◦ + 45◦ = 75◦ ) and 65 is adjacent to the
75◦ angle. Again you can use tangent.
x+y
65
◦
65 tan 75 = x + y
tan 75◦ =
x + y ≈ 242.58 f t
Since y = 65 f t, x, must equal 242.58 − 65 = 177.58 f t.
The car traveled 177.58 feet.
Karen is 5.5 feet tall and looks up at a 40◦ angle to see the top of the flagpole in front of a building. She is standing
40 feet from the flagpole. How tall is the flagpole?
Again, start by making a detailed picture of the situation and labeling what you know.
Now, make a plan for how you will solve. Look to see what is given and what you are looking for and think about
what method or technique would be helpful. In this situation, you have a right triangle. You know an angle and a
side within the right triangle (40◦ and 40 ft). You are looking for another side of the triangle (x).
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Chapter 1. Triangles in Applied Problems
Think of the flagpole as being made up of two pieces. The first piece is Karen’s height of 5.5 feet. The next piece
is the rest of the flagpole that is taller than Karen (x in the picture above). x is opposite the 40◦ angle and 40 ft is
adjacent to the 40◦ angle. This is a tangent relationship.
x
40
x = 40 tan 40◦
tan 40◦ =
x ≈ 33.56 f t
Therefore, the complete height of the flagpole is approximately 33.56 + 5.5 = 39.06 f eet.
Examples
Example 1
Earlier, you were asked to find the area and perimeter of the garden.
A community garden is being built in your neighborhood. The garden will be triangular in shape and a fence will
surround the garden. Two sides of the garden will be 33 feet and 24 feet. The angle between those two sides will
measure 62◦ . Find the area and perimeter of the garden.
Start by drawing a picture and carefully labeling everything that you know.
To find the area of the garden you can use the sine area formula.
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1
A = (33)(24) sin 62◦
2
A ≈ 349.6 f t 2
In order to find the perimeter of the garden you need to know the length of the third side. In this situation you know
two sides and an included angle, so you can use the Law of Cosines to find the length of the third side.
332 + 242 − 2(33)(24) cos 62◦ = x2
1089 + 576 − 743.64 = x2
921.36 = x2
x ≈ 30.35 f t
Now that you know the length of the third side, you can find the perimeter of the garden.
P = 33 + 24 + 30.35 = 87.35 f t
A surveyor wants to find the distance from points A and B to an inaccessible point C. These three points form a
triangle. Standing at point A, he finds m6 A in the triangle is equal to 60◦ . Standing at point B, he finds m6 B in
the triangle is equal to 55◦ . He measures the distance from point A to point B and finds it to be 350 feet. Find the
distance from points A and B to point C.
Example 2
Draw a picture of this situation.
Here is a picture:
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Chapter 1. Triangles in Applied Problems
Example 3
Make a plan: what method(s) or technique(s) can you use to solve this problem?
You can find the measure of angle C using the fact that the sum of the measures of the three angles in a triangle is
180◦ . Then, you will know all the angles and one side. You could then use the Law of Sines to set up equations to
find AC and BC.
Example 4
Solve the problem.
m6 C = 65◦ . Now, use the Law of Sines twice:
To find AC:
sin 65◦ sin 55◦
=
350
AC
350 sin 55◦
AC =
sin 65◦
AC ≈ 316.34 f t
To find BC:
sin 65◦ sin 60◦
=
350
BC
350 sin 60◦
AC =
sin 65◦
AC ≈ 334.44 f t
The distance from A to C is approximately 316 feet and the distance from B to C is approximately 334 feet.
Review
The angle of depression of a boat in the distance from the top of a lighthouse is 25◦ . The lighthouse is 200 feet tall.
Find the distance from the base of the lighthouse to the boat.
1. Draw a picture of this situation.
2. Make a plan: what method(s) or technique(s) can you use to solve this problem?
3. Solve the problem.
A pilot is flying due west and gets word that a major storm is in her path. She turns the plane 40◦ to the left of her
intended course and continues the flying. After passing the storm, she turns 50◦ to the right and flies until she has
returned to her original flight path. At this point she is 75 miles from where she left her original path when she first
made a turn. How much further did the pilot fly as a result of the detour?
4. Draw a picture of this situation.
5. Make a plan: what method(s) or technique(s) can you use to solve this problem?
6. Solve the problem.
A new bridge is being built across a river in your town. You want to figure out how long the bridge will be. You find
two points on one side of the river that are 30 feet apart. These two points with the point at the end of the bridge on
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the other side of the river form a triangle. You stand at each of the two points on your side of the river and measure
the angles of the triangle. You find the two angles are 45◦ and 70◦ . How far are each of the two points on your side
of the river from the end of the bridge on the other side of the river? How long will the bridge be?
7. Draw a picture of this situation.
8. Make a plan: what method(s) or technique(s) can you use to solve this problem?
9. Solve the problem.
∆ABC has two sides of length 12 and a non-included angle that measures 60◦ .
10. Draw a possible picture of this situation.
11. Find the measure of all sides and angles of ∆ABC.
12. Find the area of ∆ABC.
Lily starts at point A and walks straight for 100 feet. Then, she turns right at an 80◦ angle and continues walking
for another 150 feet. In order to go straight back to her starting place, how far will she need to walk? At want angle
should she turn right?
13. Draw a picture of this situation.
14. Make a plan: what method(s) or technique(s) can you use to solve this problem?
15. Solve the problem.
Answers for Review Problems
To see the Review answers, open this PDF file and look for section 7.8.
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