Trigonometric ratios in right triangles
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Trigonometric Ratios in Right Triangles. Geometry Mr. Oraze. Trigonometric Ratios are based on the Concept of Similar Triangles!. 1. 45 º. 2. 1. 1. 45 º. 2. 45 º. All 45º- 45º- 90º Triangles are Similar!. 30º. 30º. 2. 60º. 60º. 1. 30º. 60º.

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Trigonometric Ratios in Right Triangles

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Trigonometric Ratios in Right Triangles

Geometry

Mr. Oraze


Trigonometric Ratios are based on the Concept of Similar Triangles!


1

45 º

2

1

1

45 º

2

45 º

All 45º- 45º- 90º Triangles are Similar!


30º

30º

2

60º

60º

1

30º

60º

All 30º- 60º- 90ºTriangles are Similar!

4

2

1

½


All 30º- 60º- 90ºTriangles are Similar!

10

60º

2

60º

5

1

30º

30º

1

60º

30º


c a

b

The ratio is called the Tangent Ratio for angle 

The Tangent Ratio

c’ a’

b’

If two triangles are similar, then it is also true that:


Side

Opposite

q

Side

Adjacent

q

Naming Sides of Right Triangles

Hypotenuse

q


Tangent q =

Hypotenuse

Side

Opposite

q

q

Side

Adjacent

q

The Tangent Ratio

There are a total of six ratios that can be made

with the three sides. Each has a specific name.


Hypotenuse

Side

Opposite

q

q

Side

Adjacent

q

The Six Trigonometric Ratios(The SOHCAHTOA model)

S O H C A H T O A


Hypotenuse

Side

Opposite

q

q

Side

Adjacent

q

The Six Trigonometric Ratios

The Cosecant, Secant, and Cotangent of q

are the Reciprocals of

the Sine, Cosine,and Tangent of q.


2

1

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?


y

x

q

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)


y

x

q

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.


y

r

y

x

q

x

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.


y

q

x

Trigonometric Values for angles in Quadrants II, III and IV

Pick a point on the

terminal ray and raisea perpendicular

to the x-axis.


y

q

x

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


y

x

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


y

x

Signs of Trigonometric Functions

Students

All

Take

Calculus

is a good way to

remember!


y

(0, 1)

x

90º

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


1

45 º

1

For Reciprocal Ratios, use the facts:

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.)


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


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