This is because similar triangles have the same ratios. On the unit circle the base length of the triangle is cos(theta) and the height of the triangle is sin(theta) and the hyptenuse has length 1.
If you change the length of the hypotenuse to any other number and call it "r." Then the base and height will change proportionally.
The new triangle will have base length r times cos(theta) and the new height will be r times sin(theta). Now the pythagorean theorem gives you:
(rsin(theta))^2 + (rcos(theta))^2 = r^2
dividing out r^2 leads to the same identity:
sin^2 (theta) + cos^2 (theta) = 1.
In fact if you go back and solve for the base and height of the triangle given by using "r" instead of 1 you end up with the more general forms of sine and cosine:
sin(theta) = y/r
cos(theta) = x/r
where r is the radius of whatever circle you want and (x,y) is the point at which the angle touches the circle of that radius. When you use r=1 you get the original definition of sine and cosine that comes from the unit circle.
PaulFirmBreasts t1_j6en12n wrote
Reply to eli5: why dies the pythagorean identity work always if it is made with unit circle in mind? by [deleted]
This is because similar triangles have the same ratios. On the unit circle the base length of the triangle is cos(theta) and the height of the triangle is sin(theta) and the hyptenuse has length 1.
If you change the length of the hypotenuse to any other number and call it "r." Then the base and height will change proportionally.
The new triangle will have base length r times cos(theta) and the new height will be r times sin(theta). Now the pythagorean theorem gives you:
(rsin(theta))^2 + (rcos(theta))^2 = r^2
dividing out r^2 leads to the same identity:
sin^2 (theta) + cos^2 (theta) = 1.
In fact if you go back and solve for the base and height of the triangle given by using "r" instead of 1 you end up with the more general forms of sine and cosine:
sin(theta) = y/r
cos(theta) = x/r
where r is the radius of whatever circle you want and (x,y) is the point at which the angle touches the circle of that radius. When you use r=1 you get the original definition of sine and cosine that comes from the unit circle.