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: $x = 2.1588 + 2\pi n$, $x = 5.3004$
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$\begingroup$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.
You can interpret $\tan(x) = -1.5$ as telling you that $x$ is such that the ratio of $\sin(x)$ over $\cos(x)$ is $-1.5$. This means $|\sin(x)|>|\cos(x)|$ and that exactly one of $\sin(x)$ or $\cos(x)$ is negative. This gives us a couple candidates on the unit circle. Values of $x$ that satisfy these requirements can be found in the intervals $$(\pi/2+2k\pi, 3\pi/4+2k\pi) \quad \text{and} \quad (3\pi/2+2k\pi, 7\pi/4+2k\pi)$$ So we can find a solution in each interval. Notice that these intervals differ by $\pi$ radians, which means that if $x$ satisfies your equation then so does $x+\pi$, and $x+2\pi$ and $x+k\pi$ etc. One way to see what is happening is to imagine the set $\{s_1,s_2,s_3,\ldots \}$ as containing every solution to $\tan(x) = -1.5$. That is, $\tan(s_i)=-1.5$ for every integer $i$. Your book is choosing all the $s_i$'s where $i$ is even, and you chose all the $s_i$'s where $i$ is odd. Neither is wrong, and the union of your answer with the books answer gives you all the solutions. It would be better to generalize both answers as one and write the solution as $x+k\pi$ where $x$ can be $\approx 2.1588$ or $\approx -0.9820$
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