The Trigonometric
Functions we will be
looking at
SINE
COSINE
TANGENT
The Trigonometric
Functions
SINE
COSINE
TANGENT
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.
Find the sine, the cosine, and the tangent of theta.
Give a fraction.
37
35
12
q
Shrink yourself
down and stand
where the angle is.
opp
35
sin q 

hyp
37
adj
12
cos q 

hyp
37
35
opp
tan q 

adj
12
Now, figure out your ratios.
Find the sine, the cosine, and the tangent of theta
24.5
8.2
q
23.1
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
Now, figure out your ratios.
Using Trig Ratios to
Find a Missing SIDE
To find a
missing SIDE
1. Draw stick-man at the
2.
3.
4.
5.
given angle.
Identify the GIVEN sides
(Opposite, Adjacent, or
Hypotenuse).
Figure out which trig
ratio to use.
Set up the EQUATION.
Solve for the variable.
1.
H
Problems match the WS.
Where does x
reside?
A
x
If you see it cos
cos15

up high
9
then we
MULTIP 9  cos15  x
LY!
8.7  x
2.
H
Problems match the WS.
O
Where does x reside?
If X is down
below,
The X and the
angle will
switch…
9
sin 50 
x
SLIDE &
DIVIDE x 
9
sin 50
x  11.7
3.
H
Problems match the WS.
10
cos 51 
x
10
x
cos 51
A
x  15.9
Steps to finding the missing angle of a right
triangle using trigonometric ratios:
OPP
1. Redraw the figure
2.67 km
ADJ
HYP
q
and mark on it
HYP, OPP, ADJ
relative to the
unknown angle
Steps to finding the missing angle of a right
triangle using trigonometric ratios:
2. For the unknown
OPP
2.67 km
ADJ
HYP
q
angle choose the
correct trig ratio
which can be used
to set up an
equation
3. Set up the equation
Steps to finding the missing angle of a right
triangle using trigonometric ratios:
4. Solve the equation
OPP
2.67 km
ADJ
HYP
q
to find the
unknown using the
inverse of
trigonometric
ratio.
 Your turn
Practice Together:
q
3.1 km
2.1 km
Find, to one decimal
place, the unknown
angle in the triangle.
YOU DO:
4m
q
7m
Find, to 1 decimal place,
the unknown angle in
the given triangle.
Sin-Cosine
Cofunction
The Sin-Cosine Cofunction
sin q  cos(90  q)
cos q  sin(90  q)
1. What is sin A?
30 15

34 17
2. What is Cos C?
3. What is Sin Z?
24 12

26 13
4. What is Cos X?
5. Sin 28 = ?
cos62
6. Cos 10 = ?
sin80
7. ABC where B = 90.
Cos A = 3/5
What is Sin C?
3
5
8. Sin q = Cos 15
What is q?
75
Trig Application
Problems
MM2G2c: Solve application problems
using the trigonometric ratios.
Depression and Elevation
angle of depression
angle of elevation
horizontal
horizontal
9. Classify each angle as angle of
elevation or angle of depression.
Angle of Depression
Angle of Elevation
Angle of Depression
Angle of Elevation
Example 10
 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
tanq 
10,560
q  2
Example 11
 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
tan38 
x
x  32meters
Example 12
 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
tanq 
31.4
q  34
Example 13
 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
tanq 
75
q  48
Example 14
 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 nq 
935
q  44
Example 15
 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 the tenth.
x
tan71.5 
50
x  154.4feet
Example 16
 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 n45 
2.8
q  2meters
Example 17
 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
tanq 
5
q  31
Example 18
 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 
60
x  10feet