Trigonometric Ratios

Trigonometric Ratios

Trigonometry – Mrs. Turner

Hipparcus – 190 BC to 120 BC – born in Nicaea (now Turkey) was a Greek astronomer who is considered to be one of the first to use trigonometry.

Parts of a Right Triangle

Hypotenuse sideOpposite side

A

B

CNow, imagine that you move from angle A to angle B still facing into the triangle.

Imagine that you, the happy face, are standing at angle A facing into the triangle.

The hypotenuse is neither opposite or adjacent.

You would be facing the opposite side

and standing next to the adjacent side.

You would be facing the opposite side

and standing next to the adjacent side.

Opposite side

Adjacent side

YasuakiJapanese Mathematician

Review

Hypotenuse

Hypotenuse

Opposite Side

Adjacent SideA

B

For Angle A

This is the Opposite Side

This is the Adjacent Side

For Angle B

AThis is the Adjacent Side

This is the Opposite Side

Opposite Side

Adjacent Side

B

Hilda HudsonBritish Mathematician

Trig Ratios

We can use the lengths of the sides of a right triangle to form ratios. There are 6 different ratios that we can make.

Using Angle A to name the sidesUse Angle B to name the sides

The ratios are still the same as before!!

A

B

Hypotenuse

Adjacent Side

opposite

Adjacent

Opposite

Hypotenuse

Adjacent

Hypotenuse

Opposite

Opposite

Adjacent

Adjacent

Hypotenuse

Opposite

Hypotenuse

Szasz Hungarian Mathematician

Trig Ratios

• Each of the 6 ratios has a name• The names also refer to an angle

Hypotenuse

Adjacent

OppositeA

Sine of Angle A = Hypotenuse

Opposite

Cosine of Angle A = HypotenuseAdjancet

Tangent of Angle A = Adjacent

Opposite

Cosecant of Angle A = Opposite

Hypotenuse

Secant of Angle A = Adjacent

Hypotenuse

Cotangent of Angle A = Opposite

Adjacent

BirkhoffAmerican Mathematician

Trig RatiosHypotenuse

Adjacent

OppositeIf the angle changes from A to B

The way the ratios are made is the same

B

Sine of Angle = Hypotenuse

Opposite

Cosine of Angle = HypotenuseAdjancet

Tangent of Angle = Adjacent

Opposite

Cosecant of Angle = Opposite

Hypotenuse

Secant of Angle = Adjacent

Hypotenuse

Cotangent of Angle = Opposite

Adjacent

B

B B

B

B B

FreitagGerman Mathematician

Trig Ratios

• Sine, Cosine and Tangent ratios are the most common. Adjacent

OppositeA

Hypotenuse• Each of these ratios has an abbreviation

Sin A =

Cos A =

Tan A =

Csc A=

Sec A =

Cot A =

Sine of Angle A = Hypotenuse

Opposite

Cosine of Angle A = HypotenuseAdjancet

Tangent of Angle A = Adjacent

Opposite

Cosecant of Angle A = Opposite

Hypotenuse

Secant of Angle A = Adjacent

Hypotenuse

Cotangent of Angle A = Opposite

Adjacent

John DeeEnglish Mathematician

SOHCAHTOA

AdjacentA

B

OppositeHypotenuse

Here is a way to remember how to make the 3 basic Trig Ratios

1) Identify the Opposite and Adjacent sides for the appropriate angle

2) SOHCAHTOA is pronounced “Sew Caw Toe A” and it means

Sin is Opposite over Hypotenuse, Cos is Adjacent over Hypotenuse, and Tan is Opposite over Adjacent

Use the underlined letters to make the word SOH-CAH-TOA

QueteletFlemish Mathematician

Examples of Trig Ratios

6 10

8A

BFirst we will find the Sine, Cosine andTangent ratios for Angle A.

Next we will find the Sine, Cosine, andTangent ratios for Angle B Adjacent

Opposite

Remember SohCahToa

4

3

8

65

4

10

85

3

10

6

Sin A =

Cos A =

Tan A =

Sin B =

Cos B =

Tan B = 3

4

6

85

3

10

65

4

10

8

LameFrench Mathematician

Examples of Trig Ratios10

8A

BNow, we will find the Cosecant, Secant andCotangent ratios for Angle A.

Next we will find the Cosecant, Secant, andCotangent ratios for Angle B

Adjacent

Opposite

Remember SohCahToa backwards

6

Csc B =

Sec B =

Cot B = 4

3

8

63

5

6

104

5

8

10

3

4

6

84

5

8

103

5

6

10

Csc A =

Sec A =

Cot A =

BennekerAfrican American Mathematician

Special Triangles

A

B

The short side is always opposite the smaller angle.

In this triangle, angle B is smaller than angle A

Hypotenuse

Hypotenuse

A

B

In this triangle, Angle A is smaller than angle B

AlbertusGerman Mathematician

Special TrianglesThere are two special triangles: The 30-60-90 triangle and the 45-45-90 triangle.

30

60

45

45

These two right triangles are used often, so you should memorize the lengths of the sides opposite these angles.

AlbertiItalian Mathematician

30-60-90

If one of the acute angles is 30 , the %other must be 60 . %

30

60

When the side opposite the 30 angle %is 1 unit, then the side opposite the 60 angle is units and the %hypotenuse is 2 units.

3

1

2 3

NasirIslamic Mathematician

30-60-90

1

30

60

2 3

First, we will write the trigonometric ratios of the angle that measures 30 . %

Sin 30 = ̊�

Cos 30 = ̊�

Tan 30 = ̊�

2

1

2

3

3

3

3

1

Second, we will write the trigonometric ratios of the angle that measures 60 . %

Sin 60 = ̊�

Cos 60 = ̊�

Tan 60 = ̊�

2

3

2

1

31

3

Remember Soh-Cah-Toa

OleinikUkraine Mathematician

45-45-90

If one acute angle of a right triangle is 45 , then the other acute angle must %be 45 . %%

If the side opposite one 45 is 1 unit, %then the side opposite the other 45 is %also 1 unit. The hypotenuse is 2

45

45

1

1

2

CristoffelFrench Mathematician

Sin 45 = ̊�

Cos 45 = ̊�

Tan 45 = ̊�

2

2

2

1

2

2

2

1

11

1

45-45-90

45

45

1

12

First, we will write the trigonometric ratios of the angle that measures 45 . %

Since the other acute angle is also 45 , the ratios will be the same. %

BattagliniItalian Mathematician