I want some way to find the unit circle table of values without having to learn it by heart. Is there a way to do it? I thought you could for example just calculate using your calculator: $sin(0.5) = \dfrac{1}{6}\pi$ but I get a different answer..
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$\begingroup$Well, it's no wonder you get a different answer; the correct statement is $\sin(\pi/6)=1/2$ (you have just mis-remembered it as $\sin(1/2)=\pi/6$).
If you have trouble with a calculator, it may be set to "degree mode" as opposed to "radians mode".
To figure out the values that the trig functions take on the usual angles ($0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$, and $90^\circ$, or in radians, $0$, $\pi/6$, $\pi/4$, $\pi/3$, and $\pi/2$), you can just remember two facts:
- A $30\text{-}60\text{-}90$ triangle has sides in the proportions $1:\sqrt{3}:2$.
- A $45\text{-}45\text{-}90$ triangle has sides in the proportions $1:1:\sqrt{2}$.
For example, to calculate $\sin(60^\circ)$, draw the $30\text{-}60\text{-}90$ triangle and note that $$\sin(60^\circ)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{a\sqrt{3}}{2a}=\frac{\sqrt{3}}{2}.$$
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