CHAPTER 9.10
TRIGONOMETRIC
RATIOS
By: Arielle Green
Mod 9
opposite
 Sin =
hypotenuse
adjacent
 Cos =
hypotenuse
opposite
 Tan =
adjacent
VOCABULARY

Angle of Elevation – the angle between an upward
line of sight and the horizontal
A
angle of elevation
B
C
Horizontal line
ABC
is the angle of elevation.
SAMPLE PROBLEM 1

A girl was walking in the woods when she stopped
10 ft away from a tree. She spotted a birds nest at
an angle of elevation of 37˚. How far up from the
ground was the birds nest rounded to the nearest
tenth?
First choose the formula
needed for this problem.
We are working with the
two legs of the right
triangle, so we will use tan.
Set up the formula and
solve for x.
x
tan 37 
10
x  10  tan 37
x  7.5 ft
Q
X
37˚
R
10
S
VOCABULARY

Angle of Depression - the angle between a downward
line of sight and the horizontal
X
horizontal line
W

<
>
Angle of
depression
line of sight
Y
Z
WXY
is the angle of depression.
An airplane pilot is flying over a
forest at an altitude of 1600 ft.
Suddenly, he spots a fire. He
measures the angle of depression
and finds it to be 46˚. How far is the
fire, rounded to the nearest tenth,
from a point on land directly below
the plane?
There are two ways to solve this problem. We’ll
look at both ways.
A
D


46˚
Using parallel lines

alt. int.s
, <ACB is
also 46˚. Since only the
two legs of the right
triangle are being used,
the formula must be Tan
= opposite .

1600
B
46˚
X
C
adjacent
1600
Set up the equation and solve for x. Tan 46 =
x
1600
x = tan 46
x
 1545.1 ft
D
A
46˚

Since <BAC and
<CAD are
1600
complementary <s,
<BAC is 44˚. Only the
two legs of the right
triangle are being
opposite
used,
so
the formula
B
C
X
adjacent
must be Tan =
.
x
Set up the equation and solve for x. Tan 44 =
1600
44˚
X = 1600 ∙ tan 44
x  1545.1 ft
PRACTICE PROBLEMS
ROUND ALL ANSWERS TO THE NEAREST TENTH.
ROUND ALL ANGLES TO THE NEAREST DEGREE.
1.) A lighthouse casts a shadow of 55 ft when the
sun is at an angle of elevation of 67˚. How tall is
the lighthouse?
2.) A cat was on a cliff when it saw a mouse down
below at an angle of depression of 25˚. The cliff is
43 ft tall. How far away is the mouse from the
bottom of the cliff?
3.)A 25-foot ladder just reaches a point on a wall
24 ft above the ground. What is the angle of
elevation of the ladder?
PRACTICE PROBLEMS
4.)

5.)
Two men are on the opposite
sides of a tall building with
the angle of elevation being
30 and 60 respectively. If the
one man is 40 feet away from
the base of the building, how
far away is the other man?

6.)

30˚
60˚
40
x
A pole 40 ft high has a
shadow the length of 23 ft at
this point in time. Find the
angle of elevation of the sun.
Harry was walking along a
pier. He stopped when he
saw a boat on the lake at an
angle of depression of 22˚. If
the boat is 65 ft away, how
high, rounded to the nearest
tenth, is the pier from the
water ?
ANSWERS TO THE PRACTICE
PROBLEMS
1.)
x
55
x  55  tan 67
x  129.6 ft
tan 67 
x
67˚
55
2.)
65˚
x
43
x  43  tan 65
x  92.2 ft
tan 65 
43
x
3.)
24
25
 24 
x  sin
¯¹

25


x  73.7
sin x 
24
25
X˚
ANSWERS TO PRACTICE PROBLEMS (CONT’D)
4.)
5.)
40
23
40 
¯¹
x  tan


 23 
x  60.1˚
tan x 
23.094
8
30˚
60˚
40
y
tan 30 
40
y  40  tan 30
y  23.094
40
x˚
23
x
23.094
tan 60 
x
23.094
x
tan 60
x  13.3 ft
6.)
22˚
68˚
tan 68 
x
65
tan 68
x  26.3 ft
x
22˚
65
65
x
WORKS CITED
Rhoad,Richard, George Milauskas, and Robert
Whipple. Geometry for Enjoyment and
Challenge. Boston: McDougal Little, 2004.
423-427.
 “Math:Trigonometry.”Syvum. 2008. Syvum
technologies. 29 May 2008. <
http://www.syvum.com/cgi/online/serve.cgi/m
at h/trigo/trig3.sal >.
