Trigonometric Equation involving tangent - How do I solve it? I would like to know how to solve this and most importantly make intuitive sense of it. Thanks in advance!
Equation: $\tan(x) = -1.5$
(using a calculator with $\arctan(-1.5)$)
My Results: $x = -0.9820 + 2 \pi n$
Book says the result is:  $x = 2.1588 + 2\pi n$, $x = 5.3004$ 

When you have an equation $$\tan x = \tan \alpha$$ then there are an infinte number of solutions of x. The solutions such that $x\in(-\pi,\pi]$ are called the principle solutions. The general solution of the above equation is $$x=n\pi+\alpha\,;\;\;\;\;n\in\mathbb{Z}.$$ Here $\alpha = \arctan -1.5$
So the solutions are $\approx -0.9820 + n\pi$. All three solutions are correct.
You can see this from the graph. Since the the function tan x is periodic so the graph of tan x repeats after every pi units. The places where the line y=-1.5 intersects y=tan x are the solutions. As the graph keeps repeating, so there are infinite solutions. All solutions are at a distance of pi.

