If I open the MS calculator and put it in deg mode at take the sin(2.23) I get .0389. Then if I put the calc into rad mode and take the sin(.039) I get .039. If I then use google to convert .039 to degrees, I get 2.23. What is going on here?
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$\begingroup$So yeah, it's basically because of the small-angle approximation. As Angina said, this holds for angles measured in radians.
Take for example (in radians) $\sin(1) \approx 0.84$ (~$16$% difference), but $\sin(0.5) \approx 0.48$ (~$1$% difference).
So what you are doing is taking $\sin(2.23º) = 0.389$, then since $\sin(0.0389) \approx 0.0389$, you get the same number in the calculator. And then this same number you convert to degrees. It's only consistent that you get the same $2.23º$.
$\endgroup$ 5 $\begingroup$The biggest error occurs when setting 2.23 deg equal to 0.039 rads. It will be more precise if you use: $ [\operatorname{rad}] = \frac{[\deg]\pi}{180} $
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