trigonometric ratios in right triangles

21
Trigonometric Ratios in Right Triangles Geometry Mr. Oraze

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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. - PowerPoint PPT Presentation

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Page 1: Trigonometric Ratios  in Right Triangles

Trigonometric Ratios in Right Triangles

Geometry

Mr. Oraze

Page 2: Trigonometric Ratios  in Right Triangles

Trigonometric Ratios are based on the Concept of Similar Triangles!

Page 3: Trigonometric Ratios  in Right Triangles

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

45 º

2

2

22

45 º

1

1

2

45 º

1

2

1

2

1

Page 4: Trigonometric Ratios  in Right Triangles

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

1

60º

30º

½

23

32

60º

30º

2

4

2

60º

30º

1

3

Page 5: Trigonometric Ratios  in Right Triangles

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

10 60º

30º

5

35

2 60º

30º1

3

160º

30º 21

23

Page 6: Trigonometric Ratios  in Right Triangles

The Tangent Ratio

c a

b

c’ a’

b’

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

'

b

a

b

a

The ratio is called the Tangent Ratio for angle b

a

Page 7: Trigonometric Ratios  in Right Triangles

Naming Sides of Right Triangles

q

qq

Page 8: Trigonometric Ratios  in Right Triangles

The Tangent Ratio

q

qq

Tangent =q Adjacent

Opposite

There are a total of six ratios that can be madewith the three sides. Each has a specific name.

Page 9: Trigonometric Ratios  in Right Triangles

The Six Trigonometric Ratios(The SOHCAHTOA model)

q

qq

Adjacent

OppositeTangentθ

Hypotenuse

AdjacentCosineθ

Hypotenuse

OppositeSineθ

Page 10: Trigonometric Ratios  in Right Triangles

The Six Trigonometric Ratios

q

qq

Adjacent

OppositeTangentθ

Hypotenuse

AdjacentCosineθ

Hypotenuse

OppositeSineθ

Opposite

AdjacentCotangentθ

Adjacent

HypotenuseSecantθ

Opposite

HypotenuseCosecantθ

The Cosecant, Secant, and Cotangent of q are the Reciprocals of

the Sine, Cosine,and Tangent of .q

Page 11: Trigonometric Ratios  in Right Triangles

Solving a Problem withthe Tangent Ratio

60º

53 ft

h = ?

We know the angle and the side adjacent to 60º. We want to know the opposite side. Use thetangent ratio:

ft 92353

531

3

5360tan

h

h

h

adj

opp

1

2 3

Why?

Page 12: Trigonometric Ratios  in Right Triangles

Trigonometric Functions on a Rectangular Coordinate System

x

y

q

Pick a point on theterminal ray and drop a perpendicular to the x-axis.

(The Rectangular Coordinate Model)

Page 13: Trigonometric Ratios  in Right Triangles

Trigonometric Functions on a Rectangular Coordinate System

x

y

q

Pick a point on theterminal ray and drop a perpendicular to the x-axis.

ry

x

The adjacent side is xThe opposite side is yThe hypotenuse is labeled rThis is called a REFERENCE TRIANGLE.

y

x

x

yx

r

r

x

y

r

r

y

cottan

seccos

cscsin

Page 14: Trigonometric Ratios  in Right Triangles

Trigonometric Values for angles in Quadrants II, III and IV

x

yPick a point on theterminal ray and drop a perpendicular to the x-axis.

qy

x

r

y

x

x

yx

r

r

x

y

r

r

y

cottan

seccos

cscsin

Page 15: Trigonometric Ratios  in Right Triangles

Trigonometric Values for angles in Quadrants II, III and IV

x

yPick a point on theterminal ray and raise a perpendicular to the x-axis.

q

Page 16: Trigonometric Ratios  in Right Triangles

Trigonometric Values for angles in Quadrants II, III and IV

x

yPick a point on theterminal ray and raise a perpendicular to the x-axis.

y

x

x

yx

r

r

x

y

r

r

y

cottan

seccos

cscsin

qx

ry

Important! The is

ALWAYS drawn to the x-axis

Page 17: Trigonometric Ratios  in Right Triangles

Signs of Trigonometric Functions

x

y

All are positive in QI

Tan (& cot) are positive in QIII

Sin (& csc) are positive in QII

Cos (& sec) are positive in QIV

Page 18: Trigonometric Ratios  in Right Triangles

Signs of Trigonometric Functions

x

y

All

Take

Students

Calculus

is a good way toremember!

Page 19: Trigonometric Ratios  in Right Triangles

Trigonometric Values for Quadrantal Angles (0º, 90º, 180º and 270º)

x

y

90º

Pick a point one unit from the Origin.

(0, 1)

r

x = 0y = 1r = 1

090cotundefined is 90tan

undefined is 90sec090cos

190csc190sin

Page 20: Trigonometric Ratios  in Right Triangles

Trigonometric Ratios may be found by:

45 º

1

1

2Using ratios of special triangles

145cot145tan

245sec2

145cos

245csc2

145sin

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:

tan1cot

cos1sec

sin1csc

Page 21: Trigonometric Ratios  in Right Triangles

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