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Chapter 5 Introduction to
Trigonometry: 5.7 The Primary
Trigonometric Ratios
Humour Break
5.7 The Primary Trigonometric Ratios
Goals for Today:
• Learn how to identify sides and angles in
triangles
• Learn the primary trigonometric ratios
• Use the primary trigonometric ratios to find a
missing side
• Use the primary trigonometric ratios to find a
missing angle
5.7 Primary Trigonometric Ratios
In previous math course, you learned how to
calculate a missing side in a right angle
triangle when you were given two other sides.
You did this using the pythagorean theorem
a² + b² = c²
5.7 Primary Trigonometric Ratios
You also learned to how to calculate missing
angles using angle rules, such as the 180°
triangle rule, where the sum of angles in a
triangle add up to 180 °
5.7 Primary Trigonometric Ratios
Now, we are going to introduce the
trigonometric ratios, where we work with
both angles and sides to find unknown angles
in sides.
Today, we are working with the primary trig
ratios which are used for right angle triangles
only
5.7 Primary Trigonometric Ratios
What are the sides
from the perspective
of angle A?
5.7 Primary Trigonometric Ratios
From the perspective of angle A....
Hypotenuse
(or side “c”)
Opposite (or
side “a”)
Adjacent (or side “b”)
5.7 Primary Trigonometric Ratios
What are the sides
from the perspective
of angle B?
5.7 Primary Trigonometric Ratios
From the perspective of angle B....
Hypotenuse
(or side “c”)
Adjacent*
(or side “a”)
Opposite* (or side “b”)
* Side name changes because from the
perspective of a different triangle
5.7 Primary Trigonometric Ratios
For a right angle triangle, the three primary trig
ratios are:
opposite
sin A 
hypotenuse
adjacent
cos A 
hypotenuse
opposite
tan A 
adjacent
5.7 Primary Trigonometric Ratios
An acronym to help remember these formulas is
SOHCAHTOA
5.7 Primary Trigonometric Ratios
With these ratios you are dealing with one of
two situations
(1) You have two sides in a triangle and you use
them to find an angle by using the inverse or
2nd of SIN, COS, or TAN on your calculator
(2) You have a side and an angle in a triangle and
you want to find another side by using the
SIN, COS, or TAN on your calculator
5.7 Primary Trigonometric Ratios
Consider the classic 3, 4, 5 right triangle
5.7 Primary Trigonometric Ratios
Let’s use scenario 1 to find angle A. Which ratio?
5.7 Primary Trigonometric Ratios
SIN because side a = 3 (opposite) and side c = 5
(hypotenuse) from angle A
5.7 Primary Trigonometric Ratios
SIN ∠ A = opposite
hypotenuse
Opposite
Sin  A 
Hypotenuse
3
Sin  A 
5
Sin  A  0.6
 A  36.9 deg rees
Hint: Once you have the
trig ratio, you use your
SIN to -1 or inverse
function to convert the
ratio 0.6 to the angle of
36.9°
5.7 Primary Trigonometric Ratios
Your calculator does a nice job. In the old days
(when I was in high school) you had to look up
the ratio in a table and convert it into an
angle!
5.7 Primary Trigonometric Ratios
Let’s again use scenario 1 to find angle A. Which
ratio? Why did it change?
5.7 Primary Trigonometric Ratios
COS because side b = 4 (adjacent) and side c = 5
(hypotenuse) from angle A
5.7 Primary Trigonometric Ratios
COS ∠ A = adjacent
hypotenuse
Adjacent
Cos  A 
Hypotenuse
4
Cos  A 
5
Cos  A  0.8
 A  36.9 deg rees
Hint: Once you have the
trig ratio, you use your
COS to -1 or inverse
function to convert the
ratio 0.8 to the angle of
36.9°
We got the same angle
which makes sense!...
Same triangle!
5.7 Primary Trigonometric Ratios
Finally, let’s again use scenario 1 to find angle A.
Which ratio?
5.7 Primary Trigonometric Ratios
TAN because side b = 4 (adjacent) and side a = 3
(opposite) from angle A
5.7 Primary Trigonometric Ratios
TAN∠ A = opposite
adjacent
Opposite
Tan  A 
Adjacent
3
Tan  A 
4
Cos  A  0.75
 A  36.9 deg rees
Hint: Once you have the
trig ratio, you use your
TAN to -1 or inverse
function to convert the
ratio 0.75 to the angle
of 36.9°
We got the same angle
which makes sense!...
Same triangle!
5.7 Primary Trigonometric Ratios
Let’s use scenario 1 to find angle
B. Which ratio? What’s different
than when we were finding
angle A?
5.7 Primary Trigonometric Ratios
When we were finding angle A, given these sides, we
used the SIN ratio. From angle B we are dealing with the
adjacent sides and the hypotenuse so we have to use the
COS ratio.
5.7 Primary Trigonometric Ratios
COS ∠ B = adjacent
hypotenuse
Adjacent
Cos  B 
Hypotenuse
3
Cos  B 
5
Cos  B  0.6
 B  53.1deg rees
Hint: Once you have the
trig ratio, you use your
COS to -1 or inverse
function to convert the
ratio 0.6 to the angle of
53.1°
5.7 Primary Trigonometric Ratios
Let’s use scenario 1 to find angle B.
Which ratio? What’s different than
when we were finding angle A?
5.7 Primary Trigonometric Ratios
When we were finding angle A, given these sides, we used
the COS ratio. From angle B we are dealing with the
opposite sides and the hypotenuse so we have to use the
SIN ratio.
5.7 Primary Trigonometric Ratios
SIN ∠ B = opposite
hypotenuse
Opposite
Sin  B 
Hypotenuse
4
Sin  B 
5
Sin  B  0.8
 B  53.1deg rees
Hint: Once you have the
trig ratio, you use your
SIN to -1 or inverse
function to convert the
ratio 0.6 to the angle of
36.9°
5.7 Primary Trigonometric Ratios
Let’s use scenario 1 to find
angle B. Which ratio?
What’s different than when
we were finding angle A?
5.7 Primary Trigonometric Ratios
When we were finding angle A, given these sides, we used
the TAN ratio. From angle B we are dealing with the
opposite sides and the adjacent sides again, so we still use
the TAN ratio, but the numbers are reversed
5.7 Primary Trigonometric Ratios
TAN∠ A = opposite
adjacent
Opposite
Tan  B 
Adjacent
4
Tan  B 
3
Cos  B  1.3333
 B  53.1deg rees
Hint: Once you have the
trig ratio, you use your
TAN to -1 or inverse
function to convert the
ratio 1.3333 to the
angle of 53.1°
We got the same angle
which makes sense!...
Same triangle!
5.7 Primary Trigonometric Ratios
Now, lets again consider the classic 3, 4, 5 right triangle,
but this time, given an angle and a side and asked to
find a side
5.7 Primary Trigonometric Ratios
From the perspective of angle A, we are dealing with
the opposite side and the hypotenuse… so we have
to use the SIN ratio…
5.7 Primary Trigonometric Ratios
Now, lets again consider the classic 3, 4, 5 right triangle,
but this time, given angle A and side c and asked to
find a side a…
This is scenario 2… given an angle and a side…
find another side…
5.7 Primary Trigonometric Ratios
SIN ∠ A = opposite
hypotenuse
a
Sin 36.9 deg rees 
5
a
0.6004 
5
a
5 x0.6004  x5
5
3.002  a
a3
You input 36.9 into
your calculator and
hit the SIN button to
get the ratio 0.6004.
In some calculators,
the order is reversed…
5.7 Primary Trigonometric Ratios
Similiarly, if asked to side side b, from the perspective of angle A,
we are here dealing with the adjacent side side and the
hypotenuse… which ratio would we use?
5.7 Primary Trigonometric Ratios
The COS ratio because from the perspective of angle A, we are
dealing with the adjacent side and the hypotenuse…
5.7 Primary Trigonometric Ratios
COS ∠ A = adjacent
hypotenuse
b
Cos36.9 deg rees 
5
b
0.7997 
5
b
5 x0.7997  x5
5
3.9985  b
b4
You input 36.9 into
your calculator and
hit the COS button to
get the ratio 0.7997.
In some calculators,
the order is reversed…
5.7 Primary Trigonometric Ratios
Similiarly, if asked to find side b, from the perspective of angle A,
but we were given side a… we sould be dealing with the
opposite side and the adjacent side side… which ratio would
we use?
5.7 Primary Trigonometric Ratios
The Tan ratio, because we are dealing with the opposite and
adjacent sides…
5.7 Primary Trigonometric Ratios
Tan ∠ A = opposite
adjacent
3
Tan36.9 deg rees 
b
3
0.7508 
b
3
(b)( 0.7508)  ( )(b)
b
0.7508b  3
0.7508
3
b
0.7508
0.7508
b  3.9957
b4
You input 36.9 into
your calculator and
hit the TAN button to
get the ratio 0.7508.
In some calculators,
the order is reversed…
5.7 Primary Trigonometric Ratios
If we now examine things from the perspective of angle B, we
are dealing with the adjacent side and the hypotenuse… so
we have to use the COS ratio…
We are again dealing with Scenario 2… given
a side and an angle, finding another side…
5.7 Primary Trigonometric Ratios
COS ∠ B = adjacent
hypotenuse
a
Cos53.1deg rees 
5
a
0.6004 
5
a
5 x0.6004  x5
5
3.002  a
a3
You input 53.1 into your
calculator and hit the COS
button to get the ratio
0.6004. In some calculators,
the order is reversed… Note
that COS 53.1° is the same as
SIN 36.9°
5.7 Primary Trigonometric Ratios
In this next example… again from the perspective of angle B, we
are dealing with the opposite side and the hypotenuse… so
we have to use the SIN ratio…
We are again dealing with Scenario 2… given
a side and an angle, finding another side…
5.7 Primary Trigonometric Ratios
SIN ∠ B = opposite
hypotenuse
b
Sin53.1 deg rees 
5
b
0.7997 
5
b
5 x0.7997  x5
5
3.9985  b
b4
You input 53.1 into your
calculator and hit the SIN
button to get the ratio
0.7997. In some calculators,
the order is reversed…
5.7 Primary Trigonometric Ratios
In this final example… again from the perspective of angle B, we
are dealing with the opposite side and the adjacent side… so
we have to use the TAN ratio…
We are again dealing with Scenario 2… given
a side and an angle, finding another side…
5.7 Primary Trigonometric Ratios
TAN ∠ B = opposite
adjacent
b
Tan53.1 deg rees 
3
b
1.3319 
3
b
3 x1.3319  x3
3
3.9957  b
b4
You input 53.1 into your
calculator and hit the SIN
button to get the ratio
0.7997. In some calculators,
the order is reversed…
Homework
• Tuesday, December 3rd – p.496, #1, 2, 4, 9-11
• Thursday, December 12th – p.498, #12-15, 17,
19-21
• Friday, December 13th – p.498, #22-26