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Solving Problems
with Trigonometry
Connections
Have you ever . . .
• Modeled a problem using a right triangle?
• Had to find the height of a flagpole or column?
• Wondered how far away a helicopter was?
Trigonometry can be used to solve real-world situations involving
triangles. For instance, a point on the ground, the top of a
flagpole, and the bottom of a flagpole form a right triangle.
Suppose you know the height of the flagpole and the angle of
elevation from the bottom to the top of the flagpole. How can
you use trigonometry to find the distance from the point on the
ground to the bottom of the flagpole?
Trigonometry is the study of the relationships between the sides and angles of a triangle. It
can be used to find an unknown side or angle of a right triangle.
A trigonometric ratio is a fraction formed by two sides of a triangle. Three important
trigonometric ratios are sine (sin), cosine (cos) and tangent (tan). To find these ratios in a
right triangle, start with one of the acute angles, such as the one labeled using the uppercase Greek letter theta (H) in the diagram. Notice which sides are the hypotenuse, opposite,
and adjacent sides and how to calculate sine, cosine, and tangent. You can use these
definitions to find unknown side lengths of a right triangle.
sin H =
opposite
hypotenuse
adjacent
hypotenuse
tan H =
opposite
adjacent
H
opposite
cos H =
se
u
oten
p
y
h
adjacent
1
Essential Math Skills
Learn
It!
Using Trigonometry to Find a Distance
Here’s an example of a problem using trigonometry. In the
figure below, two side lengths are given, an angle measure
is given, and the side labeled x is unknown. What if you want
to find the value of x?
sin(31) = x
7
cos(31) = 6
7
tan(31) = x
6
Math Tip
You can use your
calculator to find
the sine, cosine, or
tangent of an angle.
Press the sin, cos,
or tan button, then
enter the angle
measurement, and
then press enter.
Remember: Every
right triangle has
two acute angles
and one right angle.
The longest side of
every right triangle
is the hypotenuse.
The other two sides
are called legs.
7
x
Math Tip
31c
6
You can write an equation to find the value of x. Two of the trigonometric ratios have the length x you want to find, so either one will do. Let’s
use sin(31) = x . To find the value of x, you can multiply both sides by 7.
7
sin(31) = x $ 7sin(31) = x
7
So, x is equal to 7sin(31). You can use your calculator to find x as a
decimal. Rounded to the nearest hundredth, sin(31) = 0.52, which you
can use to find x.
x = 7sin(31) = 7(0.52) = 3.64
When a right triangle represents a real-world situation, trigonometric ratios can help you to
answer questions about missing information.
A flagpole has a height of 35 feet. From a point on the ground, the angle of
elevation from the bottom of the flagpole to the top of the flagpole is 52°. What is
the distance from the point to the bottom of the flagpole?
Draw and Label a Diagram to Visualize the Problem
Drawing a diagram helps you to see how a right triangle models the problem. Use x to
represent what you want to find.
?
2
1. On a separate sheet of paper, draw and label a right triangle to represent the problem.
Solving Problems with Trigonometry
Draw a right triangle so that the bottom leg represents the ground, and the other leg
represents the height of the flagpole. Use x for the distance you want to find.
35
52c
x
Use the Diagram to Write a Trigonometric Ratio
Using a trigonometric ratio will help you to find the missing information.
?
2. What trigonometric ratio can you use to help you find x?
The sides opposite and adjacent to the 52° angle are the known side and the side you want
to find. The only trigonometric ratio that uses opposite and adjacent is tangent:
tan(52) = 35
x
Solve the Equation and Write the Solution
Solve the equation by getting the variable alone on one side of the equation.
?
3.Find x and then write the solution.
Solve the equation by multiplying both sides by x and then dividing by tan(52) to get x alone
on one side of the equation.
tan(52) = 35
x
$ x tan(52) = 35 $ x =
35
$ x = 35
tan(52)
1.28
$ x = 27.34
3
Essential Math Skills
e
ic
Pract
It!
Answer the following questions. Round all decimal
answers to the nearest hundredth.
1. You are standing on the ground 550 yards
from the bottom of a cliff. From where you
are standing, the angle of elevation from the
bottom of the cliff to the top of the cliff is 8°.
sin H =
opposite
hypotenuse
cos H =
adjacent
hypotenuse
tan H =
opposite
adjacent
a. Draw a right triangle to represent this situation.
b. Write a trigonometric ratio that can be used to find the height x of the cliff.
c. Solve your ratio to find the height, in yards, of the cliff.
2. A helicopter is hovering 1,625 feet above the ground without moving. The angle of
elevation from the landing pad to the helicopter is 22°.
4
a. Draw a right triangle to represent this situation.
b. Write a trigonometric ratio that can be used to find the distance x from the
helicopter to the landing pad.
c. Solve your ratio to find the distance in feet from the helicopter to the landing pad.
Solving Problems with Trigonometry
3. A 30-meter steel cable is connected at the top of a radio tower to a point on the
ground. From this point, the angle of elevation from the bottom of the tower to the top
of the tower is 48°.
a. Draw a right triangle to represent this situation.
b. Write a trigonometric ratio that can be used to find the distance x from the point
to the bottom of the tower.
c. Solve your ratio to find the distance in meters from the point on the ground to the
bottom of the tower.
4. Jonas wants to find the value of x this the right triangle.
70c
x
Build Your
Math Skills
16
You can use
other facts about
triangles along
with trigonometric
ratios.
a. Write a trigonometric ratio that Jonas can use to find x.
b. Solve your ratio to find x.
Remember that
the Pythagorean
theorem can help
you calculate the
length of sides.
Understanding
that the sum of the
angles in a trianlge
equals 180° can
help you calculate
unknown angles.
5
Essential Math Skills
Math Tip
5. A scuba diver is 14.5 meters below sea level. The angle of decline
from a boat to the diver is 30°.
An angle of elevation
is an angle formed
from the horizontal
line up to a point.
a. Draw a right triangle to represent this situation.
An angle of decline
is an angle formed
from the horizontal
line down to a point.
b. Write a trigonometric ratio that can be used to find the distance
x from the boat to the diver.
c. Solve your ratio to find the distance in meters from the point on the ground to the
bottom of the tower.
6. In a right triangle, one of the angles is 70° and the side opposite the 70° angle is
14 inches.
6
a. Draw a right triangle to represent this situation.
b. Write a trigonometric ratio that can be used to find the side x that is adjacent to
the 70° angle.
c. Solve your ratio to show that the length of the adjacent side is
14
inches.
tan(70)
Solving Problems with Trigonometry
Check Your Skills
Use your knowledge of right triangles and trigonometry to answer the following questions.
1.
What is the value of cos(62) rounded to the nearest hundredth?
a. 0.88
b. 0.47
c. 1.88
d. 0.62
2. For the figure below, which answer is equal to cos H?
H
17
8
15
3.
8
a. 17
15
b. 8
17
c. 8
15
d. 17
A lamp post casts a shadow 6.8 meters long. From the end of the shadow, the
angle of elevation from the bottom to the top of the lamp post is 51°.
How tall is the lamp post, rounded to the nearest tenth of a meter?
a. 8.4 m
b. 5.9 m
c. 4.3 m
d. 6.8 m
7
Essential Math Skills
4. Select all equations that are true for the figure below.
25
7
H
24
25
o sin H = 24
24
o cos H = 25
7
o tan H = 25
24
o sin H = 7
7
o cos H = 25
7
o tan H = 24
5. A wire is connected from the top of a tower to a point on the ground that is 62 feet
from the bottom of the tower. From this point, the angle of elevation from the bottom
of the tower to the top of the tower is 75°. What is the length of the wire?
62
a.
ft
tan(75)
b. 62 cos(75) ft
c. 62 tan(75) ft
62
d.
ft
cos(75)
6. For the figure below, what is the value of sin H?
3
a. 5
4
b. 5
5
c. 3
5
3
d. 8
H
3
8
Remember
the Concept
Trigonometry can be
used to solve problems
involving right triangles.
You can find an
unknown side length
using trigonometry
when you know the
measure of one of the
acute angles and the
length of another side.
Answers and Explanations
Solving Problems with Trigonometry
Using Trigonometry to Find a Distance
2c.4337.88 feet
Solve the equation.
sin(22) = 1,625
x
Practice It!
First, multiply by x.
1a.
x sin(22) = 1,625
Then, divide by sin(22).
8c
x = 1,625
sin(22)
550
The distance 550 is the distance along the ground to
the bottom of the cliff, and the smallest angle is 8°
(the angle of elevation to the top of the cliff).
x
1b. tan(8) =
550
Use x to represent the height of the cliff. Use the
tangent ratio because the height x of the cliff is
opposite the 8° angle, and the distance from you to
the bottom of the cliff (550 yards) is the adjacent leg.
Using a calculator, divide 1,625 by sin(22).
x = 4337.88
3a.
30
1c.77.30 yards
Solve the equation.
tan(8) =
48c
x
550
Multiply both sides by 550.
550 tan(8) = x
Using a calculator, multiply 550 by tan(8).
x = 77.30
2a.
1,625
22c
The distance 1,625 is the height above the ground,
so this distance and the ground make the legs of the
triangle. The elevation of 22° is the angle from the
ground toward the helicopter.
2b. sin(22) = 1,625
x
Use x to represent the distance from the landing
pad to the helicopter (the hypotenuse). Use the sine
ratio because the height (1,625 ft) of the helicopter
is opposite the 22° angle, and the distance x from the
helicopter to the landing pad is the hypotenuse.
The distance 30 would be the hypotenuse of the
triangle, because it’s the distance from the point to
the top of the tower. The angle 48° is the angle from
the point, looking up to the top of the tower.
x
3b. cos(48) =
30
Use x to represent the distance from the point to the
bottom of the tower. Use the cosine ratio because
the distance x from the point to the bottom of the
tower is the adjacent side, and the length of the cable
(30 m) is the hypotenuse.
3c.20.07 meters
Solve the equation.
cos(48) = x
30
Multiply both sides by 30:
30 cos(48) = x
Using a calculator, multiply 30 by cos(48).
x = 20.07
x
16
Use the sine ratio because the side opposite the 70°
angle is x, and the hypotenuse is 16.
4a. sin(70) =
i
Essential Math Skills
4b.15.04
6a.
Solve the equation.
sin(70) = x
16
Multiply both sides by 16:
16 sin(70) = x
Using a calculator, multiply 16 by sin(70).
14
x = 15.04
5a.
boat
70c
30c
14.5
diver
The distance 14.5 is the leg down from the surface of
the water to the diver. The angle 30° is the angle from
the boat, looking down at the diver.
5b. sin(30) = 14.5
x
Use x to represent the distance from the boat to
the diver. Use the sine ratio because the depth of
the diver (14.5 m) is opposite the 30° angle, and
the distance x from the boat to the diver is the
hypotenuse of the right triangle.
5c.29.00 meters
Solve the equation.
sin(30) = 14.5
x
Use x to represent the adjacent side. Use the tangent
ratio because the opposite side (14 in.) is given, and x
is the adjacent side.
6c.Divide by x and then divide by tan(70) to show that x
is equal to this ratio:
14
tan(70)
tan(70) = 14
x
Multiply both sides by x.
x tan(70) = 14
Divide both sides by tan(70).
x=
14
tan(70)
Multiply both sides by x.
Check Your Skills
x sin(30) = 14.5
1.
b. 0.47
Divide both sides by sin(30).
Use a calculator. Press the cos button, then 62, and
then press enter. Round the result to the nearest
hundredth (2 decimal places).
8
2.
a. 17
x = 14.5
sin(30)
Using a calculator, divide 14.5 by sin(30).
x = 29.00
ii
The triangle may be turned a different way, but the
leg that is length 14 will be opposite the angle that
is 70°.
6b. tan(70) = 14
x
The cosine ratio is the side adjacent to H (which is 8)
divided by the hypotenuse (which is 17).
Answers and Explanations
3.
a. 8.4 m
Draw a diagram using x for the height of the
lamp post.
62
ft
cos(75)
Draw a diagram and let x represent the length of
the wire.
5.
d.
x
x
51c
6.8 m
The sides opposite and adjacent to the 51° angle are
the side you want to find and the known side, which
means you use the tangent ratio:
tan(51) = x
6.8
To find x, multiply both sides by 6.8:
6.8 tan(51) = x
Using a calculator, multiply 6.8 by tan(51).
x = 8.4
4.
cos H = 24
25
tan H = 7
24
The hypotenuse is 25, the opposite side is 7, and the
adjacent side is 24. This triangle has the following
trigometric ratios:
sin H = 7
25
cos H = 24
25
tan H = 7
24
75c
62 ft
Use the cosine ratio because 62 is the adjacent side
and x is the hypotenuse:
cos(75) = 62
x
Multiply both sides by x:
x cos(75) = 62
Sove for x by dividing both sides by cos(75):
x=
62
cos(75)
4
6.
b. 5
To find sin H, you need to find the opposite
side, which is unknown. Use the Pythagorean
theorem using a for the length of the
unknown side:
a2 + 32 = 52
Calculate the exponents:
a2 + 9 = 25
Subtract 9 from each side:
a2 = 16
Take the square root to find the value of
a. The value must be positive, since it’s the
measurement of a length.
a = 16 = 4
The opposite side is 4, and the hypotenuse is
5. Therefore, sin H = 4 .
5
iii