Difference between revisions of "Proofs of trig identities"
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=Pythagorean identities= | =Pythagorean identities= | ||
− | Pythagorean identities are easy and there's no algebra involved. | + | Pythagorean identities are easy and there's no algebra involved. In fact, the name Pythagorean is a giveaway of what we should do! |
==<math>\cos^2+\sin^2=1</math>== | ==<math>\cos^2+\sin^2=1</math>== | ||
The proof here is very straightforward. We use the pythagorean theorem on <math>\triangle OAB</math> giving us <math>OA^2+AB^2=OB^2</math> or <math>\sin^2+\cos^2=1^2</math>. | The proof here is very straightforward. We use the pythagorean theorem on <math>\triangle OAB</math> giving us <math>OA^2+AB^2=OB^2</math> or <math>\sin^2+\cos^2=1^2</math>. | ||
+ | |||
+ | ==<math>\cos^2+\sin^2=1</math>== |
Revision as of 15:26, 20 January 2024
Contents
[hide]Introduction
and are easy to define. I prefer the unit circle definition as it makes these proofs easier to understand. Next, we define some other functions:
Note: I've omitted because it's unnecessary and might clog things up a little.
With a bit of ingenuity, we can create the following diagram:
We can note that the functions are correct by similar triangles.
Pythagorean identities
Pythagorean identities are easy and there's no algebra involved. In fact, the name Pythagorean is a giveaway of what we should do!
The proof here is very straightforward. We use the pythagorean theorem on giving us or .