Prove tgx = a == x = arctga + n\pi, n \in Z

[Scholar]

Please help me to prove that

\(\displaystyle tg x = a \Longleftrightarrow x = arctg a + n\pi, n \in \mathbb{Z}\)

Where to begin? I've solved a decent amount of trig equations, but the proofs are new to me..

 

[stapel]

Doesn't this just follow from the definitions of tangent and arctangent...?

If x = arctan(a), then tan(x) = a.

if x = arctan(a) plus some multiple of pi then, since the tangent wave repeats itself with a period of pi, then tan(x) = a.

Working backwards, if tan(x) = a, then the definition of arctangent gives you a primary or principal value of x = arctan(a), but you know, from the behavior of the tangent wave, that this is only one of the infintely-many such values of x.

:wink: