For example $\sin x =\frac{1}{2}$, find the angle $x$ = $\frac{\pi}{6}$
I understand that using a right-angle triangle, the ratio is $1:2$ for a sine function. but how do I evaluate the value of the angle without a calculator?
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insipidintegrator
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user307640
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Sines and cosines of special angles – Théophile Sep 06 '22 at 18:11
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There was a similar question posted recently, take a look https://math.stackexchange.com/questions/4523058/how-to-find-the-measure-of-an-angle-in-degrees-radians-from-inverse-trigonometri?noredirect=1#comment9502354_4523058 – Vasili Sep 06 '22 at 18:13
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Only a few values of any trigonometric functions correspond to "nice" angles (meaning in general, a whole number of degrees or a suitable rational number tines $\pi$ radians). So if you are given a number a priori to find its inverse sine or cosine, chances are there is no compact expression other than $\sin^{-1}$or $\cos^{-1}$ of the number. For instance, the only rational numbers whose square roots have a "nice" inverse sine or cosine are $0,\pm1/4,\pm1/2,\pm3/4,\pm1$. You won't get an elegant inverse cosine of $\sqrt{1/3}$, for example.
Of course, an example may be "rigged" to give a nice angle; if so, then there are ways to reduce the equation to something from which a rational multiple of $\pi$ radians or whole number of degrees can be read off by inspection. See this answer for a couple such cases.
Oscar Lanzi
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