Difference between revisions of "Proofs of trig identities"
(→\cos^2+\sin^2=1) |
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Line 32: | Line 32: | ||
label("B",B,dir(135+degrees(d))); | label("B",B,dir(135+degrees(d))); | ||
pair C = (1,0); | pair C = (1,0); | ||
− | label("C",C, | + | label("C",C,SE); |
pair D = (1,tan(d)); | pair D = (1,tan(d)); | ||
label("D",D,N); | label("D",D,N); | ||
Line 72: | Line 72: | ||
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>\tan^2+1=sec^2</math>== | + | ==<math>\tan^2+1=\sec^2</math>== |
− | Same story here. Applying pythagorean to <math>\triangle OCD</math> | + | Same story here. Applying pythagorean to <math>\triangle OCD</math> gives us <math>OC^2+CD^2=OD^2</math> or <math>\tan^2+1^2=\sec^2</math>. |
+ | |||
+ | ==<math>1+\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>. |
Revision as of 15:30, 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 .