TRIGONOMETRIC
RATIOS
goal: know how to set up different
trig ratios
Where to start…
θ this is the symbol for an unknown angle
measure.
It’s name is ‘Theta’.
Don’t let it scare you… it’s like ‘x’ except for
angle measure… it’s a way for us to keep
our variables understandable and
organized.
Trigonometric Ratios
Name
“say”
Abbreviation
Abbrev.
Ratio of an
angle
measure
Sine
(pronounced
“sign”)
Cosine
(pronounced
“co-sign)
tangent
(pronounced
“tan-gent)
Sin
Cos
Tan
Sinθ = opposite side cosθ = adjacent side
hypotenuse
hypotenuse
tanθ =opposite side
adjacent side
Opp
Sin 
Hyp
hypotenuse
hypotenuse
Adj
Cos 
Hyp
Opp
Tan 
Adj

adjacent
opposite
One more
time…
Here are the
ratios:
sinθ = opposite side
hypotenuse
cosθ = adjacent side
hypotenuse
tanθ =opposite side
adjacent side
We could ask for the trig functions of the angle Θ by using the definitions.

c
b
SOHCAHTOA
Θ
a
You need to pay attention to which angle you want the trig function
of so you know which side is opposite that angle and which side is
adjacent to it. The hypotenuse will always be the longest side and
will always be opposite the right angle.

Oh,
I'm
acute!
This method only applies if you have
a right triangle and is only for the
acute angles (angles less than 90°)
in the triangle.
5
4
Θ
3
So
am I!
Trigonometric Ratios
B
hypotenusec
• Let ∆ABC be a right
triangle. The sine,
the cosine, and the A
tangent of the acute
angle A are defined
as follows.
b
side adjacent to angle A
cos A =
sin A =
Side opposite A
hypotenuse
=
Side
a opposite
angle A
C
Side adjacent to A
=
hypotenuse
b
c
a
c
tan A =
Side opposite A
=
Side adjacent to A
a
b
Let’s practice…
Write the ratio for sin L
M
Sin L= _a
c
c
a
N
Write the ratio for cos L
b
Cos L = _b_
c
L
Write the ratio for tan L
Let’s switch angles:
Find the sin, cos and
tan for Angle M:
Sin M = _b_
c
Tan L = _a_
b
Cos M = _a_
c
Tan M = _b_
a
65
a
Practice Together:
x
Given each triangle, write
the ratio that could be
used to find x by
connecting the angle and
sides given.
32
b
x
YOU DO:
x
c
d
56
Given the triangle, write all
the ratios that could be
used to find x by
connecting the angle and
sides given.
Ex. 2: Finding Trig Ratios
S
sin S =
cosS =
tanS =
opposite
R
hypotenuse
adjacent
opposite
5
13 hypotenuse
hypotenuse
opposite
adjacent
T
12
adjacent
S
Ex. 2: Finding Trig Ratios—Find the sine, the
cosine, and the tangent of the indicated angle.
R
sin S =
cosS =
tanS =
opposite
hypotenuse
adjacent
hypotenuse
opposite
adjacent
R
adjacent
5
13 hypotenuse
T
12
opposite
S
Ex. 1: Finding Trig Ratios
Large Triangle
sin A =
cosA =
tanA =
Small Triangle
opposite
hypotenuse
adjacent
hypotenuse
opposite
adjacent
B
B
17
8.5
4
8
A
A
15
C
7.5
C
Ex. 3: Finding Trig Ratios—Find the sine, the
cosine, and the tangent of 45
45
sin 45=
cos 45=
tan 45=
opposite
hypotenuse
adjacent
hypotenuse
opposite
adjacent
√2
hypotenuse
1
45
1
Ex. 4: Finding Trig Ratios—Find the sine, the
cosine, and the tangent of 30
30
sin 30=
cos 30=
tan 30=
opposite
hypotenuse
adjacent
hypotenuse
opposite
adjacent
2
1
30
√3