Fundamental trigonometric equation

Posted January 6th, 2008 by Structure

We want to solve some fundamental equations:

$\sin x=a,\: -1\leq a\leq 1$

$\cos x=a,\: -1\leq a\leq 1$

$\tan x=a,\: -\infty <a<+\infty$

$\cot x=a,\: -\infty <a<+\infty$

Each of these equations has a countable set of solutions.

$\sin x=a,\: -1\leq a\leq 1;\: x_k=k\pi+(-1)^k \arcsin a$

$\cos x=a,\: -1\leq a\leq 1;\: x_k=2k\pi\pm \arccos a$

$\tan x=a,\: -\infty <a<+\infty;\: x_k=k\pi+ \arctan a$

$\cot x=a,\: -\infty <a<+\infty:\: x_k=k\pi+ \cot^{-1} a$

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