New guy and programming problem

[Thomas Klemm]

(08-28-2017 06:13 PM)Dieter Wrote:  On a 10-digit machine this already happens for 89,9995°...90,0005°.

We can use this difference-to-product formula to calculate e.g. \(1-\sin(89.9995°)=\sin(90°)-\sin(89.9995°)\):

\(\sin \theta - \sin \varphi =2\sin \left({\frac {\theta - \varphi }{2}}\right)\cos \left({\frac {\theta + \varphi }{2}}\right)\)

And then since \(\theta = \frac{\pi}{2}\) we can set \(\varepsilon = \frac {\theta - \varphi }{2}\) and use \(\cos \left({\tfrac {\pi }{2}}-\varepsilon \right)=\sin \varepsilon\) and end up with:

\(1 - \sin \varphi =2\sin^2 \varepsilon\)

Thus we set \(\theta=90\) and \(\varphi=89.9995\) and get:

90
ENTER
89.9995
-
2
÷
SIN
x²
2
×

3.807717748e-11

(Calculated using an HP-41.)

Cheers
Thomas