Trigonometric Ratios
in Right Triangles

M. Bruley

Trigonometric Ratios are based on the Concept of Similar Triangles!

All 45- 45- 90 Triangles are Similar!

45

2

2

45

1

1

45

1

All 30- 60- 90 Triangles are Similar!

1

2

4

60

30

60

30

2

60

30

1

All 30- 60- 90 Triangles are Similar!

10

60

30

5

2

60

30

1

1

60

30

The Tangent Ratio

c a

b

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

c a

b

The ratio is called the Tangent Ratio for angle

Naming Sides of Right Triangles

q

q

q

The Tangent Ratio

There are a total of six ratios that can be made

with the three sides. Each has a specific name.

q

q

q

Tangent q =

The Six Trigonometric Ratios
(The SOHCAHTOA model)

q

q

q

The Six Trigonometric Ratios

The Cosecant, Secant, and Cotangent of q

are the Reciprocals of

the Sine, Cosine,and Tangent of q.

q

q

q

Solving a Problem with
the Tangent Ratio

60

53 ft

h = ?

We know the angle and the

side adjacent to 60. We want to

know the opposite side. Use the

tangent ratio:

Why?

1

2

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)

x

y

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.

x

y

q

*

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.

x

y

q

y

x

r

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

q

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

r

y

Important! The is

ALWAYS drawn to the x-axis

x

y

q

Signs of Trigonometric Functions

All are positive in QI

Tan (& cot) are

positive in

QIII

Sin (& csc) are

positive in

QII

Cos (& sec) are

positive in

QIV

x

y

Signs of Trigonometric Functions

All

Take

Students

Calculus

is a good way to

remember!

x

y

*

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

Pick a point one unit from

the Origin.

r

x = 0

y = 1

r = 1

x

y

90

(0, 1)

Trigonometric Ratios may be found by:

Using ratios of special triangles

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

45

1

1

For Reciprocal Ratios, use the facts:

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

2

2

2

2

1

2

3

3

2

3

3

5

2

1

'

'

b

a

b

a

=

b

a

Adjacent

Opposite

Adjacent

Opposite

Tangent

Hypotenuse

Adjacent

Cosine

Hypotenuse

Opposite

Sine

=

=

=

Adjacent

Opposite

Tangent

Hypotenuse

Adjacent

Cosine

Hypotenuse

Opposite

Sine

=

=

=

Opposite

Adjacent

Cotangent

Adjacent

Hypotenuse

Secant

Opposite

Hypotenuse

Cosecant

=

=

=

ft

92

3

53

53

1

3

53

60

tan

=

=

=

=

h

h

h

adj

opp

3

y

x

x

y

x

r

r

x

y

r

r

y

=

=

=

=

=

=

q

q

q

q

q

q

cot

tan

sec

cos

csc

sin

y

x

x

y

x

r

r

x

y

r

r

y

=

=

=

=

=

=

q

q

q

q

q

q

cot

tan

sec

cos

csc

sin

0

90

cot

undefined

is

90

tan

undefined

is

90

sec

0

90

cos

1

90

csc

1

90

sin

=

=

=

=

1

45

cot

1

45

tan

2

45

sec

2

1

45

cos

2

45

csc

2

1

45

sin

=

=

=

=

=

=

q

q

q

q

q

q

tan

1

cot

cos

1

sec

sin

1

csc

=

=

=