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Given $\tan\theta = -\frac{4}{3}$, between $0\leq\theta\leq2\pi$, how can I find both values of $\theta$, with or without a calculator?

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    $\begingroup$ Use your calculator to find $\tan^{-1}(4/3)$ in radians, and then subtract the result from $\pi$ (to get the angle in Quadrant II) and from $2\pi$ (to get the angle in Quadrant IV). $\endgroup$ – user84413 Jul 19 '14 at 23:05
  • $\begingroup$ Excellent! I'm trying to understand it conceptually, why does this work? $\endgroup$ – Irongrave Jul 19 '14 at 23:13
  • $\begingroup$ @Irongrave Because $\tan^{-1}$ is the inverse function of $\tan$. And so $\tan^{-1}(\tan x)=x$. $\endgroup$ – Hakim Jul 19 '14 at 23:14
  • $\begingroup$ I get that, but how does the subtraction from $\pi$ work? $\endgroup$ – Irongrave Jul 19 '14 at 23:15
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    $\begingroup$ @Irongrave I was using the facts that the angles in Quadrants II and IV with reference angle $\theta$ are given by $\pi-\theta$ and $2\pi-\theta$. The easiest way to see this is to draw a picture of the angles in a unit circle. $\endgroup$ – user84413 Jul 19 '14 at 23:26
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Consider the two points
$(-3\mid 4)$
and $(3\mid -4)$
which both are 5 units from the origin.
These can be expressed as complex numbers in a very handy form as follows, respectively:
$5\left(-{3\over 5}+ {4\over 5}i\right)$
and
$5\left({3\over 5}-{4\over 5}i\right)$

  1. What angle has as its cosine $-{3\over 5}$and as its sine ${4\over 5}$?
  2. What angle has as its cosine ${3\over 5}$and as its sine $-{4\over 5}$?
    Bear in mind that $\cos x=\cos(2\pi-x)$
    and that
    $\sin x=\begin{cases} \sin(\pi-x),&where 0\le x\le\pi\\ \sin(3\pi-x)&where \pi\le x\le 2\pi\\ \end{cases}$
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To define $tan^{-1}$, the domain of the tangent function was restricted to $(=\pi/2, \pi/2)$. Hence, using a calculator, $tan^{-1}(-4/3)$ will give you an answer in Quadrant IV; in this case, $\approx -53.13$ degrees, to which you can add $360$ degrees and obtain the answer $306.87$ degrees. To obtain the other answer, we look in Quadrant II, and so, evaluate $306.87 - 180 = 126.87$ degrees.

Therefore, your two solutions are approximately 306.87 degrees and 126.87 degrees.

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