Thank You, Pythagoras

We return again to one of the identities of supreme importance in this field: The Pythagorean Theorem. Given a right triangle with adjacent leg x, opposite leg y, and hypotenuse r, we know that

Now divide both sides of this equation by . This yields

or

Recall from our definitions of the six trigonometric functions above that

Then we see that the identity above (based on the Pythagorean Theorem) can be rewritten in terms of trig functions as

which is one of the most celebrated and used of the trigonometric identities. Essentially, it follows directly from the Pythagorean Theorem.

(One quick notational fact is in order here. We will write for and similarly for all of the other trig functions.)

From this identity, we can build two related ``Pythagorean trig'' identities, which relate to and to . Beginning with

divide through by . This yields the following:

Simplification of this yields

using the identities we mentioned in the previous section.

Similarly, we can start with

divide through by , and obtain the identity

(It is left to the reader to go through the details of this.)

We summarize these three crucial identities here: