Difference between revisions of "Proofs of trig identities"
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Same. Pythagorean on <math>\triangle OEF</math> gives <math>OE^2+EF^2=OF^2</math> or <math>1^2+\cot^2=\csc^2</math>. | Same. Pythagorean on <math>\triangle OEF</math> gives <math>OE^2+EF^2=OF^2</math> or <math>1^2+\cot^2=\csc^2</math>. | ||
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+ | ==Conclusion== | ||
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+ | Even though with the first one and the definitions, we can make the rest from algebra, having a geometric meaning is nice when we want to know what it actually means. |
Revision as of 15:33, 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 .
Same story here. Applying pythagorean to gives us or .
Same. Pythagorean on gives or .
Conclusion
Even though with the first one and the definitions, we can make the rest from algebra, having a geometric meaning is nice when we want to know what it actually means.