Trigonometric Ratios
Lesson 12.1
HW: 12.1/1-22
Warm – up
Find the missing measures. Write all answers
in radical form.
x
30°
30
45
60
45
10
z
y
60°
y
x3 3
y3 2
z 5 3
y 5
What is a trigonometric ratio?
The relationships between the angles
and the sides of a right triangle are
expressed in terms of
TRIGONOMETRIC RATIOS.
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.
Several 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….
One more thing…
θ and 𝜶
are symbols for unknown angle
measures. Their names are ‘Theta’ and
‘Alpha’, from the Greek alphabet.
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.
The 2 other angles and the 2 other sides
adjacent
A
We will refer to the
sides in terms of
their proximity to
the angle
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.
Greek letter ‘PHI’
Adjacent side

Opposite side
opposite
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.
adjacent
B
Opposite side
Greek Letter ‘Theta’

Adjacent side
Remember we won’t use
the right angle
X
Here we
go!!!!
The Trigonometric Functions
we will be looking at
SINE
COSINE
TANGENT
The Trigonometric Functions
SINE
COSINE
TANGENT
SINE
sin
Pronounced
“sign”
COSINE
cos
Pronounced
“co-sign”
TANGENT
tan
Pronounced
“tan-gent”
The Trigonometric Ratios
So, what does this stuff mean?...
Adj Leg
Cos
Hyp
Opp Leg
Tan
Adj Leg

adjacent
opposite
opposite
Opp Leg
Sin
Hyp hypotenuse
We need a way to
remember all of
these ratios…
Some
Old
Hippie
Came
A
Hoppin’
Through
Our
Old Hippie Apartment
Old Hippie
Sin
Opp
Hyp
Cos
Adj
Hyp
Tan
Opp
Adj
SOHCAHTOA
For any right-angled triangle
Definition of Sine Ratio

opposite side
sin =
hypotenuse
For any right-angled triangle
Definition of Cosine Ratio

adjacent side
cos  =
hypotenuse
For any right-angled triangle
Definition of Tangent Ratio.

opposite side
tan  =
adjacent
Find:
Sin 16°
Tan 58°
Ex. 1: Finding Trig Ratios
Large
sin A =
cos A =
tan A =
opposite
hypotenuse
8
17
adjacent
15
hypotenuse
17
opposite
adjacent
8
Small
4
≈ 0.4706
8.5
7.5
≈ 0.8824
8.5
≈ 0.4706
≈ 0.8824
4 ≈
7.5
≈ 0.5333
15
0.5333
B
B
17
8.5
4
8
A
A
15
C
7.5
C
Trig ratios are often expressed as
decimal approximations.
Ex. 2: Finding Trig Ratios
SohCahToa
<S
opp
sin S =
hyp
adj
cos S = hyp
opp
tan S = adj
5
0.3846
≈
13
12 ≈0.9231
13
5 ≈ 0.4167
12
R
opposite
5
13 hypotenuse
T
12
adjacent
S
Finding sin, cos, and tan.
(Just writing a ratio or decimal.)
Find the sine, the cosine, and the tangent of angle A.
Give a fraction and decimal answer (round to 4 places).
9
opp

sin A 
10.8  .8333
hyp
10.8
9
A
adj
6
cos A 

hyp
10.8
 .5556
6
Shrink yourself
down and stand
where the angle is.
opp
tan A 
adj
Now, figure out your ratios.
9

6
 1.5
Find the sine, the cosine, and the tangent of angle A
Give a fraction and
decimal answer (round
to 4 decimal places).
24.5
8.2
A
23.1
Shrink yourself
down and stand
where the angle is.
8 .2
opp

sin A 

.
3347
24.5
hyp
adj
cos A 
hyp
23.1

24.5  .9429
opp
tan A 
adj
8 .2

23.1  .3550
Now, figure out your ratios.
Finding an angle.
(Figuring out which ratio to use and getting to
use the 2nd button and one of the trig buttons.)
Ex. 1: Find . Round to four decimal places.
nd
2
17.2

9
17.2
tan  
9
tan 17.2 
9
)
  62.3789
Shrink yourself down and stand where
the angle is.
Now, figure out which trig ratio you have
and set up the problem.
SohCahToa
Make sure you are in degree mode (not radians).
Ex. 2: Find . Round to three decimal places.
7

23
nd
2
7
cos  
23

cos 7
23
)
  72.281
SohCahToa
Make sure you are in degree mode (not radians).
Finding the angle measure
In the figure, find 
opposite side
sin  =
sin 

hypotenuse
=
 = 34.85
4
7
4
7
Finding the length of a side
In the figure, find y
opposite side
sin 35 =
hypotenuse
sin 35 =
y
11
11* sin35 = y
y=
6.31
y
35°
11
Finding the angle measure
In the figure, find 
cos  =
cos  = =
=
adjacent Side
3

hypotenuse
3
8
67.98
8
Finding the length of a side
In the figure, find x
adjacent side
cos 42 =
hypotenuse
cos 42 =
x=
x=
6
x
6
cos 42
8.07
6
42°
x
Finding the angle measure
In the figure, find 
3
tan  =
tan  ==
opposite side
adjacent side
3
5
 = 78.69
5

Finding the length of a side
In the figure, find z
tan 22 =
tan 22 =
z=
z
Opposite side
adjacent Side
5
z
5
tan 22
z = 12.38
5
22
Conclusion
opposite side
sin  
hypotenuse
adjacent side
cos  
hypotenuse
opposite side
tan  
adjacent side
Make Sure that
the triangle is
right-angled
SohCahToa
Solving a Problem with
the Tangent Ratio
SohCahToa
h=?
60º
53 ft
We know the angle and the
side adjacent to 60º. We want to
know the opposite side. Use the
tangent ratio:
opp h
tan 60 

adj 53
h
tan 60 
53
h  53 tan 60
h  53 3  92 ft
Note:
• The value of a trigonometric ratio depends
only on the measure of the acute angle, not
on the particular right triangle that is used to
compute the value.
Ex. 4: Finding Trig Ratios—Find the sine,
the cosine, and the tangent of 30
30
sin 30=
opposite
hypotenuse
adjacent
cos 30=
hypotenuse
opposite
tan 30=
adjacent
1
2 = 0.5
√3 ≈ 0.8660
2
1 =
√3
√3
3 ≈ 0.5774
Begin by sketching a 30-60-90
triangle. To make the calculations
simple, you can choose 1 as the length
of the shorter leg. From Theorem 9.9,
on page 551, it follows that the length
of the longer leg is √3 and the length of
the hypotenuse is 2.
2
1
30
√3
Ex: 5 Using a Calculator
• You can use a calculator to approximate the
sine, cosine, and the tangent of 74. Make
sure that your calculator is in degree mode.
The table shows some sample keystroke
sequences accepted by most calculators.
Sample keystrokes
Sample keystroke
sequences
74 sin
or sin 74
Sample calculator display
Rounded
Approximation
0.961262695
0.9613
0.275637355
0.2756
3.487414444
3.4874
ENTER
74
or
COS
COS 74
ENTER
74 TAN
or TAN
74
ENTER
Notes:
• If you look back at Examples 1-5, you
will notice that the sine or the cosine of an
acute triangles is always less than 1. The
reason is that these trigonometric ratios
involve the ratio of a leg of a right triangle
to the hypotenuse. The length of a leg or
a right triangle is always less than the
length of its hypotenuse, so the ratio of
these lengths is always less than one.