### Trigonometric Ratios

```RIANGLE
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.
Finding Trig Ratios
• A trigonometric ratio is a ratio of the
lengths of two sides of a right
triangle.
• The word trigonometry is derived
from the ancient Greek language and
means measurement of triangles.
• The three basic trigonometric ratios
are called sine, cosine, and tangent
Finding Trig Ratios
Some terminology:
• Before we can use the ratios we need to get a
few terms straight
• The hypotenuse (hyp) is the longest side of
the triangle – it never changes
• The opposite (opp) is the side directly across
from the angle you are considering
angle you are considering
A picture always helps…
• looking at the triangle in terms of angle b



(near the angle)
C
B is the opposite
(across from the angle)
C is always the
hypotenuse
b
Longest
A
B
hyp
b
Near
opp Across
But if we switch angles…
• looking at the triangle in terms of angle a

A is the opposite (across
from the angle)
C
A
a


the angle)
C is always the
hypotenuse
B
Across
hyp
Longest
opp
a
Near
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.
In a right triangle, the shorter sides are called legs and the longest side
(which is the one opposite the right angle) is called the hypotenuse
A
We’ll label them a, b, and c and the angles 
and . Trigonometric functions are defined by
taking the ratios of sides of a right triangle.
c
leg
b
First let’s look at the three basic functions.
SINE
B
leg a
COSINE
TANGENT
They are abbreviated using their first 3 letters
opposite
a
sin A 

hypotenuse c
opposite a
tan A 

b
cos A 

hypotenuse c
the trig functions of the angle B using the definitions.
A
SOHCAHTOA
c
b
a
B
opposite
b
sin B 

hypotenuse c
a
cos B 

hypotenuse c
opposite b
tan B 

A
It is important to note WHICH angle you are talking
about when you find the value of the trig function.
Let's try finding some trig functions
with some numbers. Remember that
sides of a right triangle follow the
Pythagorean Theorem so
c
5
4b
a3
sin A =
o 3

h 5
B
tan B =
o
4

3
a
SOH-CAH-TOA
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!
A
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
3
B
So
am I!
One more
time…
Here are the
ratios:
sinθ = opposite side
hypotenuse
hypotenuse
tanθ =opposite side
The nice thing is that your calculator has
a tan, sin and cos key that can save you some
work.
However, you must remember to change the
Degrees. Radians is the default setting.
Here are the steps:
•Press Mode
•Move the curser down to the 3rd line “Radians”
•Slide the curser over so “degree” is blinking
•Press 2nd Quit
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
How can we use these?
Calculating a side if you know the angle
you know an angle (25°) and its adjacent side
we want to know the opposite side
A
tan 25 
6
.47 A

1
6
 6 (.47)  A
2.80  A
b
C
A
25°
B=6
opp
tan  
Another example
• If you know an angle and its opposite side,
you can find the adjacent side.
opp
tan  
6
tan 25 
B
.47 6

1 B
C
b
A=6
25°
B
.47 B  6
.47 B 6

 12.76
.47 .47
B  12.76
How can we use it?
Suppose we want to find an angle
and we only know two side lengths
Suppose we want to find angle a
• Is side A opposite or adjacent?
the opposite
• what is side B?
b
C
A=3
a
B=4
• with opposite and adjacent we
use the…
tan ratio
opp
tan  
Lets solve it
b
C
opp
tan  
3
tan a   0.75
4
0.75
a
tan
a  36.87º
A=3
a
B=4
When the tan, sin or cos is in the
denominator, we are going to use
the reciprocal buttons.
Look above tan on the calculator
You should see TAN-1
Press 2nd TAN (.75)
Another tangent example…
•
•
•
•
we want to find angle b
B is the opposite
so we use tan
4
tan b 
3
tan b  1.33
b  53.13
opp
tan  
b
C
A=3
a
B=4
Ex. 6: Indirect Measurement
• You are measuring the height of
a Sitka spruce tree in Alaska.
You stand 45 feet from the base
of the tree. You measure the
angle of elevation from a point
on the ground to the top of the
top of the tree to be 59°. To
estimate the height of the tree,
you can write a trigonometric
ratio that involves the height h
and the known length of 45 feet.
The math
opposite
tan 59° =
Write the ratio
h
tan 59° =
Substitute values
45
45 tan 59° = h
Multiply each side by 45
45 (1.6643) ≈ h
Use a calculator or table to find tan 59°
75.9 ≈ h
Simplify
The tree is about 76 feet tall.
Ex. 7: Estimating Distance
• Escalators. The escalator at
the Wilshire/Vermont Metro
Rail Station in Los Angeles
rises 76 feet at a 30° angle.
To find the distance d a
person travels on the
escalator stairs, you can write
a trigonometric ratio that
involves the hypotenuse and
the known leg of 76 feet.
d
30°
76 ft
Now the math
opposite
sin 30° =
hypotenuse
76
sin 30° =
d
Write the ratio for
sine of 30°
30°
Substitute values.
d
d sin 30° = 76
76
d=
Divide each side by sin 30°
sin 30°
76
d=
Multiply each side by d.
0.5
d = 152
Substitute 0.5 for sin 30°
Simplify
A person travels 152 feet on the escalator stairs.
76 ft
Why do we need the sin & cos?
• We use sin and cos when we need to work
with the hypotenuse
• if you noticed, the tan formula does not have
the hypotenuse in it.
• so we need different formulas to do this work
• sin and cos are the ones!
C = 10
b
A
25°
B
Lets do sin first
• we want to find angle a
• since we have opp and hyp we use sin
5
sin a 
10
sin a  0.5
a  30
opp
sin  
hyp
C = 10
b
A=5
a
B
And one more sin example
• find the length of side A
• We have the angle and
the hyp, and we need the
opp
opp
sin  
hyp
A
sin 25 
20
A  sin 25  20
A  0.42 20
A  8.45
C = 20
b
A
25°
B
And finally cos
• We use cos when we need to work with the hyp
cos  
• so lets find angle b
hyp
4
cos b 
10
cos b  0.4
b  66.42
C = 10
b
A=4
a
a  90 - 66.42
a  23.58
B
Here is an example
• Spike wants to ride down a steel
beam
• The beam is 5m long and is leaning
against a tree at an angle of 65° to
the ground
• His friends want to find out how
high up in the air he is when he
starts so they can put add it to the
doctors report at the hospital
• How high up is he?
How do we know which formula to
use???
•
•
•
•
•
Well, what are we working with?
We have an angle
We have hyp
We need opp
With these things we will use B
the sin formula
C=5
65°
So lets calculate
opp
sin 65 
hyp
opp
sin 65 
5
opp  sin 65  5
opp  0.91 5
opp  4.53
• so Spike will have fallen 4.53m
C=5
B
65°
One last example…
• Lucretia drops her walkman
off the Leaning Tower of Pisa
when she visits Italy
• It falls to the ground 2
meters from the base of the
tower
• If the tower is at an angle of
88° to the ground, how far
did it fall?
First draw a triangle
•
•
•
•
•
What parts do we have?
We have an angle
We need the opposite
Since we are working with
the adj and opp, we will use
the tan formula
B
88°
2m
So lets calculate
opp
tan 88 
opp
tan 88 
2
opp  tan 88  2
opp  28.64  2
opp  57.27
• Lucretia’s walkman fell 57.27m
B
88°
2m
An application



You look up at an angle of 65° at the top of
a tree that is 10m away
the distance to the tree is the adjacent side
the height of the tree is the opposite side
opp
tan 65 
10
opp  10 tan 65
opp  10 2.14
opp  21.4
65°
10m
What are the steps for doing one of
these questions?
1.
2.
3.
4.
5.
6.
7.
Make a diagram if needed
Determine which angle you are working with
Label the sides you are working with
Decide which formula fits the sides
Substitute the values into the formula
Solve the equation for the unknown value
Two Triangle Problems
• Although there are two triangles, you only
need to solve one at a time
• The big thing is to analyze the system to
understand what you are being given
• Consider the following problem:
• You are standing on the roof of one building
looking at another building, and need to find
the height of both buildings.
Draw a diagram
• You can measure
the angle 40° down
to the base of other
building and up 60°
to the top as well.
You know the
distance between
the two buildings is
45m
60°
40°
45m
Break the problem into two triangles.
• The first triangle:
a
• The second triangle
• note that they share a
side 45m long
• a and b are heights!
60°
45m
40°
b
The First Triangle
• We are dealing with an angle, the opposite
• this gives us Tan
a
tan 60 
45
a  tan 60  45
a  1.73 45
a  77.94m
a
60°
45m
The second triangle
• We are dealing with an angle, the opposite and the
• this gives us Tan
b
tan 40 
45
b  tan 40  45
b  0.84 45
b  37.76m
45m
40°
b
What does it mean?
• Look at the diagram now:
• the short building is
37.76m tall
• the tall building is 77.94m
plus 37.76m tall, which
equals 115.70m tall
77.94m
60°
40°
37.76m
45m
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.
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