Finding Coordinates of Partial Arc Radius Without Sine and Cosine Functions

[Fergus]

So I think this falls under the stuff I've forgotten file. We're writting a program for the CNC lathe and I need to find x and y coordinates of a partial arc radius given the angle. This trick is the controller doesn't have sine and cosine funcitonality. I'm sure this is a basic definition thing I learned in trig in High School, unfotunately the 15 years since then have killed that memory. Help?

Thanks,
Mike

 

[uart]

You probably just need x^2 + y^2 = r^2

 

[Fergus]

I looked at that but I'm going to need more. Basically we want the user to input the angle. The Pythagorean theorum gets me one equation and two unknowns, I need to come up with another equation or be able to use something to generate the sine ratio without using the sine funciton.

 

[uart]

Sine is pretty easy to approximate.

If the angle is less that 30 degrees then the approximation,

sin(x) = x, with x in radians ( equiv to sin(x) = x*pi/180 with x in degrees),

will get you less than 5% error.

If you want better use sin(x) = x - x^3 / 6 (equiv to six(x) = x*pi/180 - (x*pi/180)^3 / 6 with x in degrees) will get you approx 1% max error if x is less than 60 degrees and better than 0.1% max error if x is less than 30 degrees.

 

[djeitnstine]

Yep wat uart said was 100% correct, and for the cosine function you may want to use cos(x) = \pi - (x*\pi/180)^{2}/2 with similar accuracy. for more accuracy just add ...+ (x*\pi/180)^{4}/24

 

[CGUE]

When evaluating the polar form of complex numbers I hate dealing with angles in the 2nd, 3rd and 4th quadrants.

 

[Fergus]

Thanks for the help! I actually just stumbled on a site showing how to use a Tayor series to estimate sine and cosine ([w__.homeschoolmath.net/teaching/sine_calculator.php) and how calculators etc. use a CORDIC algorithm to caclutate the value. Guess it wasn't actually a simple thing I forgot. Of course then I went back to one of my old math books and there it was. Thanks again!

Mike