Download Trigonometric Ratios in Right Triangles

Unit 8 – Right Triangle Trig
Trigonometric Ratios
in Right Triangles
Trigonometric Ratios are
based on the Concept of
Similar Triangles!
All 45º- 45º- 90º Triangles are Similar!
2
45 º
1
1
45 º
1
2
1
2
1
2 2
2
45 º
2
All 30º- 60º- 90º Triangles are Similar!
2 30º
3
30º
60º
4
2 3
1
60º
1
30º
60º
½
2
3
2
Trigonometric functions -- the
ratios of sides of a right triangle.

adjacent
Similar Triangles Always Have
the Same Trig Ratio Answers!
c
leg
b
SINE

leg a
COSINE
TANGENT
They are abbreviated using their first 3 letters
opposite
a
opposite a
sin  

tan  

hypotenuse c
adjacent b
adjacent
b
cos  

hypotenuse c

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

3
So
am I!

Here is a mnemonic to help you memorize
the ratios.
c
b
SOHCAHTOA

adjacent
a
opposite
b
sin  

hypotenuse c
adjacent
a
cos  

hypotenuse c
opposite b
tan  

adjacent a
It is important to note WHICH angle you are talking
about when you find the value of the trig function.

c
5
Let's try finding some trig functions
with some numbers.
4b

adjacent
a3
sin  =
o 3

h 5
tan  =
o 4

a 3
Use a mnemonic and
figure out which sides
of the triangle you
need for tangent.
sine.
How do the trig answers for  and 
relate to each other?

c
5
4b

adjacent
a3
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
opp
tan A 
adj
Now, figure out your ratios.
9

6
 1.5
The Tangent of an angle is the ratio of the
opposite side of a triangle to its adjacent side.
hypotenuse
1.9 cm
opposite
adjacent
14º
7.7 cm
1.9
Tangent 14º 
7.7

0.25
opposite
Tangent A =
adjacent
3.2 cm

24º
7.2 cm
Tangent 24º

3.2
7.2
 0.45
opposite
Tangent A =
adjacent
As an acute angle of a triangle
approaches 90º, its tangent
becomes
 infinitely large
very
large
Tan 89.9º = 573
Tan 89.99º = 5,730
etc.
very small
Since the sine and cosine functions always
have the hypotenuse as the denominator,
and since the hypotenuse is the longest side,
these two functions will always be less than 1.
opposite
Sine A =
hypotenuse
adjacent
Cosine A =
hypotenuse

Sine 89º = .9998
A
Sine 89.9º = .999998
opposite
hypotenuse
Sin α =
7.9 cm
3.2 cm
24º
Sin 24º

3 .2
7 .9
 0.41
adjacent
Cosine β =
hypotenuse
7.9 cm
46º
5.5 cm
Cos 46º

5 .5
7 .9

0.70
Ex. Solve for a missing value using a trig function.
x
tan 55 
20
20 m
55
20 tan 55  x
20
tan
55
x
)
x  28.6 m
Now, figure out
which trig ratio
you have and
set up the
problem.
Ex: 2 Find the missing side. Round to the nearest
80
tan 72 
x
x tan 72  80
tenth.
80 ft
80
x
tan 72
72
x
80

(
tan
72
)
)
=
x  26 ft
Now, figure out which trig ratio
you have and set up the problem.
Ex: 3 Find the missing side. Round to the nearest
tenth.
x
283 m
24
x
sin 24  
283
283sin 24  x
Now, figure out
which trig ratio
you have and
set up the
problem.
x  115.1 m
Ex: 4 Find the missing side. Round to the nearest
tenth.
20 ft
40
x
x
cos40  
20
20 cos40  x
x  15.3 ft
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
Now, figure out which trig ratio you have
and set up the problem.
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
Make sure you are in degree mode (not radians).
)
Ex. 3: Find . Round to three decimal places.

200
sin  
400
200
nd
2
sin
200

400 )
  30
Make sure you are in degree mode (not radians).
When we are trying to find a side
we use sin, cos, or tan.
When we are trying to find an
angle
we use sin-1, cos-1, or tan-1.
A plane takes off from an airport an an angle of 18º and
a speed of 240 mph. Continuing at this speed and angle,
what is the altitude of the plane after 1 minute?
After 60 sec., at 240 mph, the plane
has traveled 4 miles
x
4
18º
SohCahToa
Soh
Sine A =
opposite
hypotenuse

x
opposite
x
Sine 18 =
4
x
0.3090 =
4
1
x = 1.236 miles

or
4
6,526 feet
hypotenuse

18º
An explorer is standing 14.3 miles from the base of
Mount Everest below its highest peak. His angle of
elevation to the peak is 21º. What is the number of feet
from the base of Mount Everest to its peak?
x
x
Tan 21 =
0.3839 =
14.3 1
14.3
x
x = 5.49 miles
= 29,000 feet


14.3
21º
A swimmer sees the top of a lighthouse on the
edge of shore at an 18º angle. The lighthouse is
150 feet high. What is the number of feet from the
swimmer to the shore?
150
150
Tan 18 =
x
0.3249 =150
x
1


x
0.3249x = 150
0.3249 0.3249
X = 461.7 ft
18º
A dragon sits atop a castle 60 feet high. An archer
stands 120 feet from the point on the ground directly
below the dragon. At what angle does the archer
need to aim his arrow to slay the dragon?
60
Tan x =
120
Tan x = 0.5
Tan-1(0.5) = 26.6º

60
x
120
Solving a Problem with
the Tangent Ratio
We know the angle and the
side adjacent to 60º. We want to
know the opposite side. Use the
tangent ratio:
h=?
2
3
60º
53 ft
1
tan 60 
opp h

adj 53
3 h
Why?

1 53
h  53 3  92 ft
Ex.
A surveyor is standing 50 feet from the base of
a large tree. The surveyor measures the
angle of elevation to the top of the tree as
71.5°. How tall is the tree?
tan
71.5°
?
50
71.5
°
Opp

Hyp
y

50
tan
71.5°
y = 50 (tan 71.5°)
y = 50 (2.98868)
y  149.4 ft
Ex. 5
A person is 200 yards from a river. Rather than
walk directly to the river, the person walks along a
straight path to the river’s edge at a 60° angle.
How far must the person walk to reach the river’s
edge?
cos 60°
x (cos 60°) = 200
200
60°
x
x
X = 400 yards
Trigonometric Functions on a
Rectangular Coordinate System
y
Pick a point on the
terminal ray and drop a
perpendicular to the x-axis.
r
y

x
x
The adjacent side is x
The opposite side is y
The hypotenuse is labeled r
This is called a
REFERENCE TRIANGLE.
y
r
x
cos  
r
y
tan  
x
sin  
r
y
r
sec  
x
x
cot  
y
csc  
Trigonometric Ratios may be found by:
Using ratios of special triangles
2
45 º
1
1
1
2
1
cos 45 
2
tan 45  1
sin 45 
For angles other than 45º, 30º, 60º you will need to use a
calculator. (Set it in Degree Mode for now.)