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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

30º

30º

2

60º

60º

1

30º

60º

4

2

1

½

10

60º

2

60º

5

1

30º

30º

1

60º

30º

c a

b

The ratio is called the Tangent Ratio for angle

c’ a’

b’

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

Side

Opposite

q

Side

Adjacent

q

Hypotenuse

q

Tangent q =

Hypotenuse

Side

Opposite

q

q

Side

Adjacent

q

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

S O H C A H T O A

Hypotenuse

Side

Opposite

q

q

Side

Adjacent

q

The Cosecant, Secant, and Cotangent of q

are the Reciprocals of

the Sine, Cosine,and Tangent of q.

2

1

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

Pick a point on the

terminal ray and drop a perpendicular to the x-axis.

(The Rectangular Coordinate Model)

y

x

q

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

Pick a point on the

terminal ray and drop a perpendicular

to the x-axis.

y

q

x

Pick a point on the

terminal ray and raisea perpendicular

to the x-axis.

y

q

x

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

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

Students

All

Take

Calculus

is a good way to

remember!

y

(0, 1)

x

90º

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:

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.)

- 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