Download 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.
Yasuaki
Japanese Mathematician
Parts of a Right Triangle
B
Hypotenuse side
Opposite side
C
A
Adjacent side
Imagine that you, the happy face, are
standing at angle A facing into the
triangle.
Now, imagine that you move from
angle A to angle B still facing into the
triangle.
You would be facing the opposite
side
You would be facing the opposite
side
and standing next to the adjacent
and standing next to the adjacent
side.
side.
The hypotenuse is neither opposite or adjacent.
Hilda Hudson
British Mathematician
Review
B
For Angle A
Hypotenuse
This is the Opposite Side
Opposite Side
This is the Adjacent Side
A
Adjacent Side
For Angle B
B
Hypotenuse
This is the Opposite Side
This is the Adjacent Side
A
Szasz
Hungarian Mathematician
Trig Ratios
B
Using
Angle
A name
to name
sides
Use
Angle
B to
thethe
sides
We can use the lengths of the sides of a right
triangle to form ratios. There are 6
different ratios that we can make.
Hypotenuse
opposite
Adjacent Side
The ratios are still the same as before!!
Opposite
Hypotenuse
Adjacent
Hypotenuse
Opposite
Adjacent
Hypotenuse
Opposite
Hypotenuse
Adjacent
Adjacent
Opposite
A
Birkhoff
American Mathematician
Trig Ratios
• Each of the 6 ratios has a name
• The names also refer to an angle
Hypotenuse
Opposite
A
Adjacent
Sine of Angle A =
Opposite
Hypotenuse
Cosecant of Angle A =
Cosine of Angle A =
Adjancet
Hypotenuse
Secant of Angle A =
Hypotenuse
Adjacent
Cotangent of Angle A =
Adjacent
Opposite
Opposite
Tangent of Angle A =
Adjacent
Hypotenuse
Opposite
Freitag
German Mathematician
Trig Ratios
B
If the angle changes from A to B
The way the ratios are made is the same
Hypotenuse
Opposite
Adjacent
Sine of AngleB =
Opposite
Hypotenuse
Cosecant of Angle B =
Cosine of Angle B =
Adjancet
Hypotenuse
Secant of Angle B =
Hypotenuse
Adjacent
Cotangent of Angle B =
Adjacent
Opposite
Opposite
Tangent of Angle B =
Adjacent
Hypotenuse
Opposite
John Dee
English Mathematician
•
•
Trig Ratios
Each of these ratios has an abbreviation
Sine, Cosine and Tangent ratios are
the most common.
Sine of Angle
Sin A =
Opposite
Hypotenuse
Hypotenuse
Opposite
A
Adjacent
Cosecant of Angle
Csc A=A =
Hypotenuse
Opposite
Cos A =A =
Cosine of Angle
Hypotenuse
Sec A =
Secant of Angle
Hypotenuse
Adjacent
Opposite
Tangent ofTan
Angle
A = A = Adjacent
A =A =
Cotangent of Cot
Angle
Adjacent
Opposite
Adjancet
Quetelet
Flemish Mathematician
SOHCAHTOA
B
Here is a way to remember how to
make the 3 basic Trig Ratios
Hypotenuse
Opposite
A
Adjacent
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
Lame
French Mathematician
Examples of Trig Ratios
First we will find the Sine, Cosine and
Tangent ratios for Angle A.
Next we will find the Sine, Cosine, and
Tangent ratios for Angle B
B
6
Opposite
10
8
Adjacent
Remember SohCahToa
6 3

Sin A =
10 5
8 4
Cos A =

10 5
6 3
Tan A = 
8 4
8 4

10 5
6 3
Cos B =

10 5
8 4
Tan B = 
6 3
Sin B =
A
Benneker
African American Mathematician
Examples of Trig Ratios
Now, we will find the Cosecant, Secant and
Cotangent ratios for Angle A.
Next we will find the Cosecant, Secant, and
Cotangent ratios for Angle B
B
10
6
Opposite
A
8
Adjacent
Remember SohCahToa backwards
10 5

6 3
10 5
Sec A =

8 4
8 4
Cot A =

6 3
Csc A =
Csc B =
Sec B =
Cot B =
10 5

8 4
10 5

6 3
6 3

8 4
Albertus
German Mathematician
Special Triangles
The short side is always opposite the
smaller angle.
B
Hypotenuse
In this triangle, angle B is smaller than
angle A
A
B
In this triangle, Angle A is smaller
than angle B
Hypotenuse
A
Alberti
Italian Mathematician
Special Triangles
There are two special triangles: The
30-60-90 triangle and the 45-45-90
triangle.
These two right triangles are used often,
so you should memorize the lengths of
the sides opposite these angles.
30
45
45
60
Nasir
Islamic Mathematician
30-60-90
30
If one of the acute angles is 30 ̊, the
other must be 60 ̊.
3
2
When the side opposite the 30 ̊ angle
is 1 unit, then the side opposite the
60 ̊ angle is 3 units and the
hypotenuse is 2 units.
60
1
Oleinik
Ukraine Mathematician
30-60-90
First, we will write the trigonometric ratios
of the angle that measures 30 ̊.
30
2
Second, we will write the
trigonometric ratios of the angle
that measures 60 ̊.
60
1
Remember Soh-Cah-Toa
Sin 30 ̊ =
Cos 30 ̊ =
1
2
3
2
1
3

Tan 30 ̊ =
3
3
3
Sin 60 ̊ =
3
2
1
Cos 60 ̊ =
2
Tan 60 ̊ =
3
 3
1
Cristoffel
French Mathematician
45-45-90
If one acute angle of a right triangle is
45 ̊, then the other acute angle must
be 45 ̊.̊
45
2
1
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
1
Battaglini
Italian Mathematician
45-45-90
45
2
First, we will write the trigonometric ratios
of the angle that measures 45 ̊.
1
45
1
Since the other acute angle is also
45 ̊, the ratios will be the same.
Sin 45 ̊ =
1
2

2
2
Cos 45 ̊ =
1
2

2
2
Tan 45 ̊ =
1
1
1