Download 6.2 Trigonometric Applications

6.2
Trigonometric
Applications
Objectives:
1.
2.
Solve triangles using trigonometric ratios.
Solve applications using triangles.
A.
Find the side x of the right triangle given below:
SOH-CAH-TOA
Since they two sides being used
on this problem are the adjacent
& the hypotenuse, then cosine
will be used. Make sure your
calculator is in degree mode.
hyp
opp
adj
Ex. #1
cos50 x

1
10
x  10 cos50
x  6.4
Find a Side
of a Triangle
B.
Find the side x of the right triangle given below:
SOH-CAH-TOA
hyp
opp
adj
Ex. #1
Since the opposite and
adjacent are being used,
tangent is chosen.
tan 22 15

1
x
x tan 22  15
15
x
tan 22
x  37.1
Find a Side
of a Triangle
C.
Find the side x of the right triangle given below:
SOH-CAH-TOA
hyp
opp
Since the opposite and
hypotenuse are being used,
sine is chosen.
sin 58 x

1
24
x  24 sin 58
x  20.4
Ex. #1
adj
Find a Side
of a Triangle
A.
Find the measure of angle θ in the triangle below:
SOH-CAH-TOA
Since all sides are given, ANY trig ratio
can be used to solve the problem.
sin  
hyp
adj
opp
Ex. #2
5
13
cos 
12
13
5
sin 1sin    sin 1 
 13 
 12 
cos1cos   cos1 
 13 
5
tan  
12
  22.6
5
tan 1tan    tan 1 
 12 
Find an Angle
of a Triangle
B.
Find the measure of angle θ in the triangle below:
SOH-CAH-TOA
Since the adjacent and hypotenuse
are given, cosine will be used.
hyp
opp
17
cos 
32
1
cos
adj
Ex. #2
1  17 
cos   cos
  57.9
Find an Angle
of a Triangle
 
 32 
C.
Find the measure of angle θ in the triangle below:
SOH-CAH-TOA
opp
adj
hyp
Since the opposite & adjacent
are given, tangent is chosen.
12
tan  
5
tan
1
tan    tan
  67.4
Ex. #2
Find an Angle
of a Triangle
1  12 
 
5

Solve the right triangle shown below:
SOH-CAH-TOA
There are 3 things to solve for on this problem.
Sides a and b, & angle θ.
To find θ we simply subtract the other two
angles from 180°.
opp
θ = 180° − 90° − 20° = 70°
hyp
adj
Ex. #3
Solving a
Right Triangle

Solve the right triangle shown below:
SOH-CAH-TOA
To find a, we will use the opposite and
the hypotenuse, so sine is chosen.
To find b, we will use the adjacent and
the hypotenuse, so cosine is chosen.
opp
hyp
adj
Ex. #3
a
b
cos 20 
sin 20 
101
101
a  101sin 20 a  101 cos 20
b  94.9
a  34.5
Solving a
Right Triangle

Solve the right triangle shown below:
SOH-CAH-TOA
Since the two legs are the same
length, then this triangle is isosceles
and the angles of θ and β are
congruent and equal to 45°.
hyp
There are four easy ways to find
the hypotenuse c:
adj
opp
Ex. #4
1.
2.
3.
4.
Trig with sine
Trig with cosine
Pythagorean Theorem
45°-45°-90° Special Right
Triangle Rule
Solving a
Right Triangle

Solve the right triangle shown below:
SOH-CAH-TOA
hyp
adj
opp
Ex. #4
8
c
8  c sin 45
8
c
sin 45
c  11.3
8
c
8  c cos 45
8
c
cos 45
c  11.3
82  82  c 2
45  45  90
64  64  c 2
x, x, x 2
128  c 2
x 8
c  128  8 2
c  11.3
c 8 2
sin 45 
cos 45 
c  11.3
Solving a
Right Triangle

A wheelchair ramp is 6 feet in length and makes a 4°
angle with the ground. How many inches does the ramp
rise off the ground?
opp
x
hyp
6
4°
adj
x
sin 4 
6
x  6 sin 4
x  0.4 ft  5 in
Since the opposite and hypotenuse
are being used, sine will be chosen.
Ex. #5
Application

A diagonal path through a rectangular park is 600 ft.
long. One side of the park measures 350 ft. long.
A. How long is the other side of the park?
B. What angle does the diagonal path make with the side
you found in question A?
x
x 2  350 2  600 2
θ
350
600
x  122,500  360,000
2
x 2  237 ,500
x  237,500
x  487.3 ft
Ex. #6
350
600
 350 
  sin 1

 600 
  35.7
sin  
Application

The angle of elevation from a point on the street to the
top of a building is 53°. The building is 60 ft. high. How
far is the point on the street from the foot of the
building?
60
53°
x
Ex. #7
60
tan 53 
x
x tan 53  60
60
x
tan 53
x  45.2
Angle of Elevation
& Depression

From the top of a 60 ft lighthouse, built on a cliff 40 ft.
above sea level, the angle of depression to a sailboat
adrift on the water is 55°. How far from the base of the
cliff is the sailboat?
The angle of depression is equal to the
angle of elevation. Additionally to form
the right triangle we must add the height
of the cliff to that of the lighthouse.
55°
60 ft
40 ft
55°
x
Ex. #8
100 ft
100
x
x tan 55  100
100
x
tan 55
x  70 ft
tan 55 
Angle of Elevation
& Depression

While on a nature walk, a person spots a small oak
tree with an angle of elevation of 25° to the top of the
tree and an angle of depression of 15° to the bottom
of the tree from eye level. The eye level is 165 cm.
A. How far is the person
standing from the tree?
165
tan 15 
x
25°
x tan 15  165
165
x  165 cm
tan 15
x  616 cm
Ex. #9
x
15°
165 cm
Angle of Elevation
& Depression

While on a nature walk, a person spots a small oak
tree with an angle of elevation of 25° to the top of the
tree and an angle of depression of 15° to the bottom
of the tree from eye level. The eye level is 165 cm.
B. How tall is the tree?
y
287 + 165 = 452 cm
x
y  x tan 25
165
25°
x
x
tan 15
15°
165
y
 tan 25
tan 15
y  287 cm
tan 25 
Ex. #9
y
165 cm
Angle of Elevation
& Depression