Trigonometric
Ratios
A RATIO is a comparison
of two numbers. For
example;
boys to girls
cats : dogs
right : wrong.
In Trigonometry, the
comparison is between
sides of a triangle.
We need to do some
housekeeping before we
can proceed…
In trigonometry, the ratio we are talking
about is the comparison of the sides of a
RIGHT TRIANGLE.
Two things MUST BE understood:
1. This is the hypotenuse.. This
will ALWAYS be the hypotenuse
2. This is 90°… this makes the
right triangle a right triangle…. Without
it, we can not do this trig… we WILL NOT
use it in our calculations because we
COULD NOT do calculations without it.
Now that we agree about the hypotenuse and right
angle, there are only 4 things left; the 2 other
angles and the 2 other sides.
A
We will refer to the sides
in terms of their proximity
to the angle
hypotenuse
adjacent
opposite
If we look at angle A, there is
one side that is adjacent to it
and the other side is opposite
from it, and of course we have
the hypotenuse.
If we look at angle B, there is
one side that is adjacent to it
and the other side is opposite
from it, and of course we have
the hypotenuse.
hypotenuse
opposite
adjacent
B
Remember we won’t
use the right angle
X
One more thing…
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.
Here we
go!!!!
Trigonometric Ratios
Name
“say”
Abbreviation
Abbrev.
Ratio of an
angle
measure
Sine
Cosine
tangent
Sin
Cos
Tan
Sinθ = opposite side cosθ = adjacent side
hypotenuse
hypotenuse
tanθ =opposite side
adjacent side
One more
time…
Here are the
ratios:
sinθ = opposite side
hypotenuse
cosθ = adjacent side
hypotenuse
tanθ =opposite side
adjacent side
Let’s practice…
Write the ratio for sin A
B
Sin A = a
c
c
Write the ratio for cos A
a
C
b
Cos A = b
c
A
Write the ratio for tan A
Let’s switch angles:
Find the sin, cos and
tan for Angle B:
Sin B = b
c
Tan A = a
b
Cos B = a
c
Tan B = b
a
Make sure you have a calculator…
Given
Ratio of sides
Angle, side
Looking for
Angle measure
Missing side
Use
SIN-1
COS-1
TAN-1
SIN, COS, TAN
Set your calculator to ‘Degree’…..
MODE (next to 2nd button)
Degree (third line down… highlight it)
2nd
Quit
Practice some more…
Find tan A:
Tan A = opp/adj = 12/21
24.19
A
12
Tan A = .5714
21
Find tan A:
Tan A = 8/4 = 2
8
4
A
Using trig ratios in equations
Remember back in 1st grade when you had
to solve:
(6)12 = x (6)
What did you do?
6
72 = x
Remember back in 3rd grade when x was in
the denominator?
(x)12 = 6 (x)
What did you do?
x
__
__
12x = 6
x = 1/2
Ask yourself:
In relation to the angle,
what pieces do I have?
34°
15 cm
Opposite and hypotenuse
Ask yourself:
x cm
What trig ratio uses
Opposite and Hypotenuse?
SINE
Set up the equation and solve:
(15) Sin 34 = x (15)
15
(15)Sin 34 = x
8.39 cm = x
Ask yourself:
In relation to the angle,
what pieces do I have?
53°
12 cm
Opposite and adjacent
x cm
Ask yourself:
What trig ratio uses
Opposite and adjacent?
tangent
Set up the equation and solve:
(12)Tan 53 = x (12)
12
(12)tan 53 = x
15.92 cm = x
x cm
Ask yourself:
In relation to the angle,
what pieces do I have?
Adjacent and hypotenuse
68°
18 cm
Ask yourself:
What trig ratio uses
adjacent and hypotnuse?
cosine
Set up the equation and solve:
(x) Cos 68 = 18 (x)
x
(x)Cos
18
_____68 =_____
cos 68 cos 68
X = 18
X = 48.05 cm
cos 68
Ok… we’ve found side lengths, now
let’s find angle measures.
Refer to your table… what function will
we use to find angle measures?
SIN-1
COS-1
TAN-1
These are
called
INVERSE
FUNCTIONS
42 cm
22 cm
θ
This time, you’re looking for theta.
Ask yourself:
In relation to the angle, what pieces
do I have? Opposite and hypotenuse
Ask yourself:
What trig ratio uses opposite
and hypotenuse? sine
Set up the equation (remember you’re looking for theta):
Sin θ = 22
42
Remember to use the inverse function
when you find theta
Sin -1 22 = θ
42
31.59°= θ
Let’s practice…
Find an angle that has a
tangent (ratio) of 2
C
2cm
B
3
Round your answer to the
nearest degree.
3cm
A
Process:
I want to find an ANGLE
I was given the sides (ratio)
Tangent is opp
adj
TAN-1(2/3) = 34°
You’re still looking for theta.
θ
Ask yourself:
22 cm
17 cm
What trig ratio uses the parts I
was given? tangent
Set it up, solve it, tell me what you get.
tan θ = 17
22
tan -1 17 = θ
22
37.69°= θ
Your assignment
Trig
Worksheets
(Kuta)