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Trigonometric Ratios
in Right Triangles
M. Bruley
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
45 º
2
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
All 30º- 60º- 90º Triangles are Similar!
2
60º
10
1
30º
30º
3
5 3
1
60º
30º
3
2
1
2
60º
5
The Tangent Ratio
c
c’
a

b
a’

b’
a a'

If two triangles are similar, then it is also true that:
b b'
The ratio a is called the Tangent Ratio for angle 
b
Naming Sides of Right Triangles



The Tangent Ratio
Tangent   Opposite


Adjacent

There are a total of six ratios that can be made
with the three sides. Each has a specific name.
The Six Trigonometric Ratios
(The SOHCAHTOA model)



Opposite
Sine θ 
Hypotenuse
Adjacent
Cosine θ 
Hypotenuse
Opposite
Tangent θ 
Adjacent
The Six Trigonometric Ratios
Opposite
Hypotenuse
Adjacent
Cosine θ 
Hypotenuse
Opposite
Tangent θ 
Adjacent
Sine θ 



Hypotenuse
Opposite
Hypotenuse
Secantθ 
Adjacent
Adjacent
Cotangent θ 
Opposite
Cosecantθ 
The Cosecant, Secant, and Cotangent of 
are the Reciprocals of
the Sine, Cosine,and Tangent of .
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
Trigonometric Functions on a
Rectangular Coordinate System
y
Pick a point on the
terminal ray and drop a
perpendicular to the x-axis.

(The Rectangular Coordinate Model)
x
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 Values for angles in
Quadrants II, III and IV
y
r
y

x
y
r
x
cos  
r
y
tan  
x
sin  
r
y
r
sec  
x
x
cot  
y
csc  
Pick a point on the
terminal ray and drop a perpendicular
to the x-axis.
x
Trigonometric Values for angles in
Quadrants II, III and IV
y
Pick a point on the
terminal ray and raise a perpendicular
to the x-axis.

x
Trigonometric Values for angles in
Quadrants II, III and IV
y
Pick a point on the
terminal ray and raise a perpendicular
to the x-axis.

y
r
x
cos  
r
y
tan  
x
sin  
r
y
r
sec  
x
x
cot  
y
csc  
x
x
r
y
Important! The  is
ALWAYS drawn to the x-axis
Signs of Trigonometric Functions
Sin (& csc) are
positive in
QII
y
All are positive in QI
x
Tan (& cot) are
positive in
QIII
Cos (& sec) are
positive in
QIV
Signs of Trigonometric Functions
y
Students
All
x
Take
Calculus
is a good way to
remember!
Trigonometric Values for Quadrantal
Angles (0º, 90º, 180º and 270º)
y
Pick a point one unit from
the Origin.
x=0
y=1
r =1
(0, 1)
r
90º
x
sin 90  1
csc 90  1
cos 90  0
tan 90 is undefined
sec 90 is undefined
cot 90  0
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 
csc 45  2
sec 45  2
cot 45  1
For angles other than 45º, 30º, 60º or Quadrantal angles, you will
need to use a calculator. (Set it in Degree Mode for now.)
For Reciprocal Ratios, use the facts:
csc   1 sin 
sec   1 cos
cot   1 tan 
Acknowledgements
 This presentation was made possible by
training and equipment provided by an
Access to Technology grant from Merced
College.
 Thank you to Marguerite Smith for the model.
 Textbooks consulted were:
 Trigonometry Fourth Edition by Larson & Hostetler
 Analytic Trigonometry with Applications Seventh
Edition by Barnett, Ziegler & Byleen