Transcript Trigonometric Ratios in Right Triangles

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
Cofunctions p. 287
There are three pairs of cofunctions:
`The sine and the cosine
The secant and the cosecant
The tangent and the cotangent
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