TITLE: How to find sin of any fraction-angles, and how do you find them in fraction forms and not in decimal forms?
QUESTION [8 upvotes]: Ok so on doing a whole lot of Geometry Problems, since I am weak at Trigonometry, I am now focused on $2$ main questions :-
$1)$ How to calculate the $\sin,\cos,\tan$ of any angle?
Some Information :- This site :- https://www.intmath.com/blog/mathematics/how-do-you-find-exact-values-for-the-sine-of-all-angles-6212 , produces a clear understanding and a detailed approach of finding the $\sin$ of any angle from $1$ to $90^\circ$ , and I found it very interesting. But now the Questions arise :-
Can you find the $\sin$, $\cos$ or $\tan$ of any fraction angles, like $39.67$? 
Can you find the $\sin$, $\cos$ or $\tan$ of recurring fractions like $\frac{47}{9}$? 
Can you find the $\sin$, $\cos$ or $\tan$ of irrationals, like $\sqrt{2}?$
Since I am a bit new to Trigonometry, I will be asking if there is a formula to find the $\sin$ of fractions, or even recurring fractions. I can use the calculator to find them obviously, but I have another Question :-
$2)$ How to calculate the trigonometric ratios of every angle in fractional form?
We all know $\sin 45^\circ = \frac{1}{\sqrt{2}}$ , but what will be $\sin 46^\circ$ in fractions? I can use a calculator to calculate the decimal of it, but it is hard to deduce the fraction out of the value, especially because the decimal will be irrational. I know how to convert recurring decimals to fractions, but this is not the case. Right now I am focused on a particular problem, which asks me to find the $\sin$ of a recurring fraction, in a fraction form. I am struggling to do this unless I clear up the ideas.
Edit: My problem is to find the $\sin$ of $\frac{143}{3}^\circ$ . I do not have any specific formula to find this, and I am mainly stuck here. I need a formula which shows how this can be done.
Can anyone help me? Thank You.

REPLY [2 votes]: I do not catch exactly what is your problem, so  I am proposing some considerations which might be useful,
at least to substantiate what you need.
a) The sine of an angle will be rational when the angle corresponds to that of an integral triangle (i.e. a Pythagorean triple).
Among the angles which are rational multiples of $\pi$, only $0, \pi /6 , \pi /2 (+ k \pi)$ provide a rational value of the sine.
see this reference.
And if the angle is a rational multiple of $\pi$ it is as well a rational multiple of $180^{\circ}$ and thus a rational value in degrees.

Therefore the sines you are looking for are irrational.

b) The methods for finding a rational approximation of a (irrational) $\sin x$ are various, and are the subject
of the hard work of many scholars of the past, when the trigonometric tables were much of need
and the computer was far to come.
An example is the Bhaskara sine approximation.
$$
\sin x^ \circ   \approx
 {{4x\left( {180 - x} \right)} \over {40500 - x\left( {180 - x} \right)}}
$$
which for one of the angles you cite as example will give
$$
\eqalign{
  & x = \left( {{{143} \over 3}} \right)^{\, \circ }
  = \left( {{{143} \over {540}}} \right)\pi \;rad\quad  \Rightarrow   \cr 
  &  \Rightarrow \quad \sin x \approx {{227084} \over {307729}} \to err = 0.17\,\%  \cr} 
$$
And when the computer arrived, also there has been much work to find suitable algorithms to express the trig functions  which are summarized in this Wikipedia article .
There it is stated that the sine of rational multiples of $\pi$ are in fact algebraic numbers (generally of degree $2$ and higher).
A recent paper illustrates a rational approximation for the $\tan$ and for $\sin , \cos$ functions.
c) However most of the algorithms developed for being implemented on computers relies on the angle expressed in radians,
and therefore they will provide a rational approximation of the sine when the angle is expressed as a rational multiple of a radian.
The problem that you pose concerns instead the sine of an angle expressed as a fraction of degrees, which is irrational in radians.
Aside from the Bhaskara's formula above,  with the other available methods you cannot avoid to introduce
a rational approximation for $\pi$, which is going to fix the threshold on the precision that can be achieved.
To this end we would better fix some lower/upper couple of bounds on $\pi$ such as
$$
\left( {{{25} \over 8},{{22} \over 7}} \right),\left( {{{91} \over {29}},{{22} \over 7}} \right),
 \cdots ,\left( {{{688} \over {219}},{{355} \over {113}}} \right), \cdots ,
\left( {{{9918} \over {3157}},{{355} \over {113}}} \right), \cdots 
$$
obtainable by a Stern-Brocot approximation.
Then if you are using the Taylor series, for instance, you have
$$
x - {1 \over 6}x^{\,3}  < \sin x < x - {1 \over 6}x^{\,3}  + {1 \over {120}}x^{\,5} 
$$
so that for the angle already considered
$$
x = \left( {{{143} \over 3}} \right)^{\, \circ }  = \left( {{{143} \over {540}}} \right)\pi \;rad
$$
and using for $\pi$ the second couple of values above, you get
$$
\left( {{{143} \over {540}}{{91} \over {29}}} \right)
 - {1 \over 6}\left( {{{143} \over {540}}{{91} \over {29}}} \right)^{\,3}
  < \sin x <
 \left( {{{143} \over {540}}{{22} \over 7}} \right)
 - {1 \over 6}\left( {{{143} \over {540}}{{22} \over 7}} \right)^{\,3}
  + {1 \over {120}}\left( {{{143} \over {540}}{{22} \over 7}} \right)^{\,5} 
$$
i.e.
$$
{{16943907583603} \over {23042336976000}} < \sin x < {{2140128005530465093} \over {2893944959388000000}}
$$
which put into decimal is
$$
0.735338 \ldots  < 0.739239 \ldots  < 0.7395132 \ldots 
$$
corresponding to a relative error of
$$
 - \,0.5\,\% \,,\; + 0.04\,\% 
$$
Of course , depending on the accuracy required, it is possible to increase the precision on $\pi$, increase the degree of the
series, or also to change to Padè approximants or other techniques.