Trigonometric Ratios in Right Triangles

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Trigonometric Ratios in Right Triangles

• M. Bruley

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Trigonometric Ratios are based on the Concept of
Similar Triangles!
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All 45º- 45º- 90º Triangles are Similar!
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All 30º- 60º- 90º Triangles are Similar!
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2
1
½
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All 30º- 60º- 90º Triangles are Similar!
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60º
2
60º
5
1
30º
30º
1
60º
30º
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The Tangent Ratio
c a
?
?
b
If two triangles are similar, then it is also
true that
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Naming Sides of Right Triangles
Hypotenuse
q
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The Tangent Ratio
There are a total of six ratios that can be
made with the three sides. Each has a specific
name.
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The Six Trigonometric Ratios(The SOHCAHTOA model)
S O H C A H T O A
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The Six Trigonometric Ratios
The Cosecant, Secant, and Cotangent of q are the
Reciprocals of the Sine, Cosine,and Tangent of q.
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Solving a Problem withthe Tangent Ratio
We know the angle and the side adjacent to 60º.
We want to know the opposite side. Use
the tangent ratio
h ?
60º
53 ft
Why?
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Trigonometric Functions on a Rectangular
Coordinate System
Pick a point on the terminal ray and drop a
perpendicular to the x-axis.
(The Rectangular Coordinate Model)
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Trigonometric Functions on a Rectangular
Coordinate System
Pick a point on the terminal ray and drop a
perpendicular to the x-axis.
r
y
x
The adjacent side is x The opposite side is y The
hypotenuse is labeled r This is called a
REFERENCE TRIANGLE.
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Trigonometric Values for angles in Quadrants II,
III and IV
Pick a point on the terminal ray and drop a
perpendicular to the x-axis.
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Trigonometric Values for angles in Quadrants II,
III and IV
Pick a point on the terminal ray and raise a
perpendicular to the x-axis.
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Trigonometric Values for angles in Quadrants II,
III and IV
Pick a point on the terminal ray and raise a
perpendicular to the x-axis.
x
y
r
Important! The ? is ALWAYS drawn to the x-axis
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Signs of Trigonometric Functions
Sin ( csc) are positive in QII
All are positive in QI
Tan ( cot) are positive in QIII
Cos ( sec) are positive in QIV
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Signs of Trigonometric Functions
All
Students
Take
Calculus
is a good way to remember!
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Trigonometric Values for Quadrantal Angles (0º,
90º, 180º and 270º)
x 0 y 1 r 1
Pick a point one unit from the Origin.
r
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Trigonometric Ratios may be found by
Using ratios of special triangles
For angles other than 45º, 30º, 60º or Quadrantal
angles, you will need to use a calculator. (Set
it in Degree Mode for now.)
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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