Difference between revisions of "Proofs of trig identities"
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label("$\cot \theta$",shift(dir(270)/4)*brace(E,O),S); | label("$\cot \theta$",shift(dir(270)/4)*brace(E,O),S); | ||
draw(shift(dir(d+90)/24)*brace(O,D)); | draw(shift(dir(d+90)/24)*brace(O,D)); | ||
− | label("$\ | + | label("$\sec \theta$",shift(dir(degrees(d)+90)/24)*brace(O,D),dir(degrees(d)+90)); |
draw(shift(dir(270)/4)*brace(E,O)); | draw(shift(dir(270)/4)*brace(E,O)); | ||
label("1",shift(dir(270)/24)*brace(C,O),S); | label("1",shift(dir(270)/24)*brace(C,O),S); | ||
Line 57: | Line 57: | ||
label("$\tan \theta$",C--D); | label("$\tan \theta$",C--D); | ||
draw(shift(dir(degrees(d)+90)/4)*brace(O,F)); | draw(shift(dir(degrees(d)+90)/4)*brace(O,F)); | ||
− | label("$\ | + | label("$\csc \theta$",shift(dir(degrees(d)+90)/4)*brace(O,F),dir(degrees(d)+90)); |
draw(shift(dir(degrees(d)+90)/4)*O--shift(dir(degrees(d)+90)/24)*O); | draw(shift(dir(degrees(d)+90)/4)*O--shift(dir(degrees(d)+90)/24)*O); | ||
draw(shift(dir(degrees(d)+90)/4)*F--shift(dir(degrees(d)+90)/24)*F); | draw(shift(dir(degrees(d)+90)/4)*F--shift(dir(degrees(d)+90)/24)*F); |
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 .