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Without a calculator, how can I calculate the sine of an angle, for example 32(without drawing a triangle)?

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    Convert $32^{\circ}$ to radian then, use Taylor expansion in the nearest known sine, say $30^{\circ} = \pi/6$. – Shuhao Cao Jun 19 '13 at 15:42
  • maybe, we can predict approximate values, from the nearest known value. that is, $sin(30\circ) $ is 0.5. probably we can guess 0.52 – dajoker Jun 19 '13 at 15:42
  • @Vijay Raghavan: But how does a calculator find the exact value? –  Jun 19 '13 at 15:47
  • @Runemoro This is how C does it: http://sourceware.org/git/?p=glibc.git;a=blob;f=sysdeps/ieee754/dbl-64/s_sin.c;hb=HEAD involving some iterative process. – Shuhao Cao Jun 19 '13 at 15:49

2 Answers2

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You can use first order approximation $\sin(x+h)=\sin(x)+\sin'(x)h=\sin(x)+\cos(x)h$

where $x$ is the point nearest to $x+h$ at which you already know the value of the $\sin$ function and its derivative $\cos$ function too.

Like for $\sin(32^0)=\sin(30^0)+\cos(30^0)*(\frac{\pi}{90})$

Here you need to take $h$ in radians which is $\frac{\pi}{90}$ for $(32^0-30^0)=2^0$

Aang
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Using Taylor expansion you get an approximation up to the desired precision

Edoardo Lanari
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  • wikipedia for the lazy – The Chaz 2.0 Jun 19 '13 at 15:50
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    I don't know why OP accepts this answer, but apparently MacLaurin series approximation doesn't work well when the angle is large, say $438748327^{\circ}$. Avatar's answer is much better than this one in terms of MSE standards. – Shuhao Cao Jun 19 '13 at 15:55
  • What about $2\pi$-periodicity? The expansion is for radiant value. – Edoardo Lanari Jun 19 '13 at 15:57
  • @Lano My point is, expansion using terms $x - x^3/3! + \cdots$ doesn't work well unless $x\ll 1$. Otherwise you have to wait $(n+1)!$ to majorize $x^{n+1}$ in the error term, which normally results computation of many terms. If you use known sine values like $\sin(\pi/24)$, you could use first order approximation(two terms in Taylor) like in Avatar's answer. – Shuhao Cao Jun 19 '13 at 16:01
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    @ShuhaoCao I think Taylor's series is actually better answer. We could always mod the angle to 360 so it should not be the problem – Thaina Sep 24 '20 at 04:32