Hi !

Question

Here is my question:

The pythagorean theorem gives the relationships between the sides of a right triangle. The following identities show the relationships between the trigonometric functions of a particular angle.

sin^2(theta) + cos^2(theta) = 1

tan^2(theta) + 1 = sec^2(theta)

1 + cot^2(theta) = csc^2(theta)

Why are these referred to the pythagorean identities?

Answer 1

Make a sketch of a right-angle triangle with hypotenuse of 1, base side (the horizontal) x, and vertical side y , call the base angle Ø

by Pythagoras:
x^2 + y^2 = 1^2 = 1

isn't cosØ = x/1 = x
then x^2 = cos^2
and isn't sinØ = y/1 = 1
then y^2 = sin^2 Ø

so in x^2 + y^2 = 1
sin^2 Ø + cos^2 Ø = 1
so we proven this important relationship and since it was base on the Pythagorean Theorem we call it a Pythagorean identity.

the other two are obtained in a very simple way

for the 1st, divide each term by cos^2 Ø

sin^2 Ø/cos^2 Ø + cos^2 Ø = 1/cos^2 Ø
tan^2 Ø + 1 = sec^2 Ø

the last is obtained by dividing each term of the original by sin^2 Ø
You try it.

Answer 2

Thank you sir