Download The Pythagorean Identity

The Pythagorean Identity
First, we will look at calculator use and mathematical notation.
sin 2 A is pronounced sine squared A and it means sin A  sin A .
Recall that sin A is the value of the sine ratio for angle A ....it does NOT mean sin A .
So , for calculation purposes, sin 2 A  (sin A)2 .
First, we will use our calculators to do a few examples......
Calculate: sin 2 45  cos2 45
Use your calculator and follow the example.....make sure you calculator is on degree setting!!!!!!!
Old School Calculators.....
Press open bracket (....then 45.....then sin......then close bracket )......then press the x 2 button.....
then the + button.....then open bracket (.....then 45......then cos.....then close bracket ).....then
the x 2 button....then =. Your answer should be 1.
DAL calculators ......
Press open bracket (.....then sin....then 45....then close bracket ).....then the x 2 button...
Then the + button.....then open bracket (......then cos....then 45.....then close bracket ).....then
the x 2 button....then =. Your answer should be 1.
Try a few more angles .....use big values, negative values......decimal values...you should always get 1.
Next we will look at a general proof......using variables and mathematical definitions.
There is an infinite number of numbers so an infinite amount of time is not enough to prove this
identity for every existing number. What follows is a formal proof.
For basic trigonometry there is our Soh Cah Toa definition using opposite, adjacent and hypotenuse.
Let’s look at a point definition for some point ( x, y).
Here is a diagram............a point on the set of x and y axes.
y
( x, y)
x
Let’s form a triangle using the point. Label the sides with x, y & r for the angle A.
(hypotenuse)
r
y (opposite)
A
x (adjacent)
sin A 
opposite
y

hypotenuse r
cos A 
adjacent
x

hypotenuse r
&
x2  y 2  r 2 .
We want to prove that sin 2 A  cos2 A  1. This is known as The Pythagorean Identity.
Proof
2
 y  x
sin A  cos A      
r r
2
2
2

y 2 x2

r2 r2

y 2  x2
r2

x2  y 2
r2

r2
r2
 1.
We have shown that for any angle A associated with a point ( x, y) that sin 2 A  cos2 A  1.