Convert trigonometric function to irrational fraction for example $\cos(\frac\pi6)$ is $\frac{\sqrt3}{2}$. How can I convert any other trigonometric  function into this type of fraction and preferably without a calculator?
 A: To find such values, generally one first learns the very popular trigonometric table:


After that, using the trigonometric identities such as:
$\rightarrow \sin θ = \frac 1{\operatorname{cosec} θ}$ or $\operatorname{cosec} θ = \frac 1{\sin θ}$ $\qquad$ (Reciprocal)
$\rightarrow \sin^2 θ + \cos^2 θ = 1$ $\qquad$ (Pythagorean)
$\rightarrow \tan θ = \frac {\sin θ}{\cos θ}$ $\qquad$ (Ratio)
$\rightarrow \sin (-θ)=-\sin θ$ $\qquad$ (Opposite Angles)
$\rightarrow \sin (90°-θ)=\cos θ$ $\qquad$ (Complementary Angles)
$\rightarrow \sin (180°-θ)=\sin θ$ $\qquad$ (Supplementary Angles)
$\rightarrow \sin(α+β)=\sin(α)\cos(β)+\cos(α)\sin(β)$ $\qquad$ (Sum & Difference)
$\rightarrow \sin 2θ = 2\sinθ \cosθ$ $\qquad$ (Double Angle)
$\rightarrow \sin (\frac θ2) = ±\sqrt {\frac {1 – \cosθ}2}$ $\qquad$ (Half Angle)
$\rightarrow \sin α + \sin β = 2 \sin\frac {α+β}2 \cos\frac {α-β}2$ $\qquad$ (Product-Sum)
$\rightarrow \sin α\sin β = \frac {\cos (α – β) – \cos (α + β)}2$ $\qquad$ (Products)
and similarly for $\cos$ and $\tan$, etc.

One can find other trigonometric functions' values in following manner:
$$\mbox{$\bf Example$: $\sin(75°)=?$}$$
Using identity: $\sin(α+β)=\sin(α)\cos(β)+\cos(α)\sin(β)$
We have, $\sin(75°)=\sin(30°)\cos(45°)+\cos(30°)\sin(45°)$
Now looking up the table:
$\sin(75°)=\frac 12.\frac 1 {\sqrt{2}} +\frac{\sqrt 3}2.\frac 1 {\sqrt{2}}$
$$\mbox{$\bf Thus,$ $\sin(75°)=\frac{\sqrt 3+1}{2\sqrt 2}$}$$

Like with everything else, application of this method takes time to get acquainted with and practice is required so that identities may come to mind at once. Also note that this method is good for certain values and not for others.
A: By Niven's theorem, the only rational multiples of $\pi$ whose sine or cosine is rational are where the sine or cosine is $0$, $\pm 1/2$ or $\pm 1$.
Now note that $\sin^2(\theta) = (1 - \cos(2\theta))/2$.  Thus if $\theta$ is a rational multiple of $\pi$ and $\sin(\theta)$ is the square root of a rational number, $\cos(2\theta)$ must be rational and $2\theta$ is a rational multiple of $\pi$, so $\cos(2\theta)$ is $0$, $\pm 1/2$ or $\pm 1$, leading to $\sin^2(\theta) = 1/2$, $1/4$, $3/4$,$0$ or $1$.  Thus the only cases where $\theta$ is a rational multiple of $\pi$ and $\sin(\theta)$ is an irrational square root of a rational number are when $\sin(\theta) = \pm 1/\sqrt{2}$ or $\pm\sqrt{3}/2$.  Similarly for $\cos$.
