Trig Ratios and
Cofunction
Relationships
Trig Ratios
SOH-CAH-TOA
SINE
Pronounced
“sign”
COSINE
Pronounced
“co-sign”
TANGENT
Pronounced
“tan-gent”
Greek Letter q
Pronounced
“theta”
Represents an unknown angle
Opp Leg
Sinq 
Hyp
hypotenuse
Adj Leg
Cosq 
Hyp
Opp Leg
tanq 
Adj Leg
q
adjacent
opposite
opposite
Finding sin, cos, and tan.
Just writing a ratio.
1. Find the sine, the cosine, and the tangent of theta.
Give a fraction.
H
O
37
35
A
12
q
Shrink yourself down and
stand where the angle is.
Identify your hypotenuse,
adjacent side, and opposite side.
opp
35
sin q 

hyp
37
adj
12
cos q 

hyp
37
35
opp
tan q 

adj
12
2. Find the sine, the cosine, and the tangent of theta
O
8.2
24.5
H
q
23.1
A
Shrink yourself down and stand
where the angle is.
8.2
opp

sin q 
24.5
hyp
adj
cos q 
hyp
23.1

24.5
opp
8.2
tan q 

adj
23.1
Identify your hypotenuse, adjacent
side, and opposite side.
Sin-Cosine
Cofunction
The Sin-Cosine Cofunction
sin q  cos(90  q)
cos q  sin(90  q)
7. Sin 28 = ?
cos62
8. Cos 10 = ?
sin80
What is Sin Z?
24 12

26 13
What is Cos X?
What is sin A?
30 15

34 17
What is Cos C?
9. ABC where B = 90.
Cos A = 3/5
What is Sin C?
3
5
10. Sin q = Cos 15
What is q?
75
Draw ABC where BAC = 90
and sin B = 3/5
11. What is the length of AB? 4
12. What is tan C? 4/3
13. Draw stick-man standing where the
angle is and mark each given side.
Then tell which trig ratio you have.
sin
O
H
C
2
1. sin A 
5
2. cos A  21
5
2
2 21

3. tan A 
21
21
5
A
2
M
4. If C = 20º, then cos C is equal to:
A. sin 70
B. cos 70
C. tan 70
Using Trig to Find
Missing Angles
and Missing Sides
Finding a missing angle.
(Figuring out which ratio to use and an
inverse trig button.)
Ex: 1 Figure out which ratio to use. Find x. Round
to the nearest tenth.
O
20
tan q  
20 m
40
1  20 
tan    x
 40 
q  26.6
o
A
40 m
x
Shrink yourself down and
stand where the angle is.
Identify the given sides as
H, O, or A.
What trig ratio is this?
Ex: 2 Figure out which ratio to use. Find x. Round
to the nearest tenth.
O
15
sin q  
15 m
50
1  15 
sin    x
 50 
q  17.5
o
H
50 m
x
Shrink yourself down and
stand where the angle is.
Identify the given sides as
H, O, or A.
What trig ratio is this?
Ex. 3: Find q. Round to the nearest degree.
17.2
tan q 
9
O
17.2
q
9
A
 17.2 
tan 
q

 9 
1
q  62
Ex. 4: Find q. Round to the nearest degree.
A
q
7
23
H
7
cos q 
23
 7 
cos    q
 23 
1
q  72
Ex. 5: Find q. Round to the nearest degree.
q
200
O
H
200
sin q 
400
 200 
sin 

q

 400 
1
q  30
Finding a missing side.
(Figuring out which ratio to use and
getting to use a trig button.)
Ex: 6 Figure out which ratio to use. Find x. Round
to the nearest tenth.
x
tan 55 
20
A
20 m
55
20 tan 55  x
x  28.6 m
O
x
Ex: 7 Find the missing side. Round to the nearest
tenth.
O
80 ft
72
x
A
80
tan 72 
x
x tan 72  80
80
x
tan 72
x  26 ft
Ex: 8 Find the missing side. Round to the nearest
tenth.
O
x
H
283 m
24
x
sin 24  
283
283sin 24  x
x  115.1 m
Ex: 9 Find the missing side. Round to the nearest
H
20 ft
tenth.
40
x
A
x
cos40  
20
20 cos40  x
x  15.3 ft
When we are trying to find a
side we use sin, cos, or tan.
When we are trying to find
an angle we use (INVERSE)
-1
-1
-1
sin , cos , or tan .
Trig Application
Problems
MM2G2c: Solve application problems
using the trigonometric ratios.
Depression and Elevation
angle of depression
angle of elevation
horizontal
horizontal
1. Classify each angle as angle of
elevation or angle of depression.
Angle of Depression
Angle of Elevation
Angle of Depression
Angle of Elevation
Example 2
• Over 2 miles (horizontal), a road
rises 300 feet (vertical). What is
the angle of elevation to the
nearest degree? 5280 feet – 1
mile
300
tan q 
10,560
q  2
Example 3
• The angle of depression from the top
of a tower to a boulder on the ground
is 38º. If the tower is 25m high, how far
from the base of the tower is the
boulder? Round to the nearest whole
number.
25
tan 38 
x
x  32meters
Example 4
• Find the angle of elevation to the top of a
tree for an observer who is 31.4 meters from
the tree if the observer’s eye is 1.8 meters
above the ground and the tree is 23.2 meters
tall. Round to the nearest degree.
21.4
tan q 
31.4
q  34
Example 5
• A 75 foot building casts an 82 foot
shadow. What is the angle that
the sun hits the building? Round to
the nearest degree.
82
tan q 
75
q  48
Example 6
• A boat is sailing and spots a shipwreck
650 feet below the water. A diver
jumps from the boat and swims 935
feet to reach the wreck. What is the
angle of depression from the boat to
the shipwreck, to the nearest degree?
650
si n q 
935
q  44
Example 7
• A 5ft tall bird watcher is standing 50
feet from the base of a large tree. The
person measures the angle of
elevation to a bird on top of the tree as
71.5°. How tall is the tree? Round to
x
the tenth.
tan71.5 
50
x  154.4feet
Example 8
• A block slides down a 45 slope for a
total of 2.8 meters. What is the change
in the height of the block? Round to
the nearest tenth.
x
si n 45 
2.8
x  2meters
Example 9
• A projectile has an initial horizontal
velocity of 5 meters/second and an
initial vertical velocity of 3
meters/second upward. At what
angle was the projectile fired, to the
nearest degree?
3
tan q 
5
q  31
Example 10
• A construction worker leans his ladder
against a building making a 60o angle
with the ground. If his ladder is 20 feet
long, how far away is the base of the
ladder from the building? Round to
the nearest tenth.
x
cos60 
20
x  10feet