Trigonometric Ratios in Right Triangles M. Bruley

Trigonometric Ratios in Right Triangles

M. Bruley

Trigonometric Ratios are based on the Concept of Similar 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

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

1

60º

30º

½

23

32

60º

30º

2

4

2

60º

30º

1

3

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

10 60º

30º

5

35

2 60º

30º1

3

160º

30º 21

23

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

Naming Sides of Right Triangles

The Tangent Ratio

Tangent Tangent Adjacent

Opposite

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

The Six Trigonometric Ratios(The SOHCAHTOA model)

Adjacent

OppositeTangentθ

Hypotenuse

AdjacentCosineθ

Hypotenuse

OppositeSineθ

The Six Trigonometric Ratios

Adjacent

OppositeTangentθ

Hypotenuse

AdjacentCosineθ

Hypotenuse

OppositeSineθ

Opposite

AdjacentCotangentθ

Adjacent

HypotenuseSecantθ

Opposite

HypotenuseCosecantθ

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

the Sine, Cosine,and Tangent of the Sine, Cosine,and Tangent of

Solving a Problem withthe Tangent Ratio

60º

53 ft

h = ?

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

ft 92353

531

3

5360tan

h

h

h

adj

opp

1

2 3

Why?

Trigonometric Functions on a Rectangular Coordinate System

x

y

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

(The Rectangular Coordinate Model)(The Rectangular Coordinate Model)

Trigonometric Functions on a Rectangular Coordinate System

x

y

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

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.

y

x

r

y

x

x

yx

r

r

x

y

r

r

y

cottan

seccos

cscsin

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.

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

x

ry

Important! The is

ALWAYS drawn to the x-axis

Signs of Trigonometric Functions

x

y

AAll are positive in QI

TTan (& cot) are positive in QIII

SSin (& csc) are positive in QII

CCos (& sec) are positive in QIV

Signs of Trigonometric Functions

x

y

AAll

TTake

SStudents

CCalculus

is a good way toremember!

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

x

y

º

Pick a point one unit from the Origin.

(0, 1)

r

x = 0y = 1r = 1

090cotundefined is 90tan

undefined is 90sec090cos

190csc190sin

Trigonometric Ratios may be found by:

45 º

1

1

2Using ratios of special trianglesUsing ratios of special triangles

145cot145tan

245sec2

145cos

245csc2

145sin

For angles other than 45º, 30º, 60º or Quadrantal angles, you will For angles other than 45º, 30º, 60º or Quadrantal angles, you will need to use a calculator. (Set it in Degree Mode for now.)need to use a calculator. (Set it in Degree Mode for now.)

For Reciprocal Ratios, use the facts:For Reciprocal Ratios, use the facts:

tan1cot

cos1sec

sin1csc

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