Holt Geometry 8-2 Trigonometric Ratios 8-2 Trigonometric Ratios Holt Geometry.
Trigonometric Ratios
-
Upload
liliana1993 -
Category
Education
-
view
10.104 -
download
2
description
Transcript of 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