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I am working through "Basic Mathematics" by Serge Lang, and there is an example in the polar coordinates section that seems incomplete to me.

Example. Find polar coordinates for the point whose rectangular coordinates are $(1,\sqrt {3})$.

Their solution is

We have $x = 1$ and $y = \sqrt{3}$, so that

$$r = \sqrt{1+3} = 2$$ Also $$cos \theta = \frac{1}{2}$$ and $$sin \theta = \frac{\sqrt{3}}{2}$$ We see that $\theta = \pi /3 $

At this point in the book, inverse sine and cosine have not been introduced. Was $\theta = \pi /3$ just an observation, or is there a method to determine the angles of a right triangle just using the side measurements without the inverse trig functions?

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  • $\begingroup$ Why to use complicated advanced methods when simple observations works ? $\endgroup$ Jan 12, 2020 at 16:44
  • $\begingroup$ There are tables available, where you can read the angle if you know the $\sin$, $\cos$, or $\tan$ value of that angle. $\endgroup$
    – YNK
    Jan 12, 2020 at 16:49

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No, there isn't. More simply, $\theta=\pi/3$ or $\theta=\pi/4$ or $\theta=\pi/6$ are values for which $\sin$ and $\cos$ are mnemonical remembered. Obiuosvly, all the values of $\sin$ and $\cos$ are real so there are expression for every value of $\theta$. The expression can be very complicated or even not known.

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