Finding Coordinates of Partial Arc Radius Without Sine and Cosine Functions

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SUMMARY

This discussion focuses on calculating the x and y coordinates of a partial arc radius without using sine and cosine functions, specifically for a CNC lathe application. The user, Mike, seeks to approximate sine and cosine values using mathematical techniques due to the controller's limitations. Key approximations include using sin(x) = x for angles less than 30 degrees and sin(x) = x - x³/6 for angles up to 60 degrees, achieving maximum errors of 1% and 0.1%, respectively. The conversation also highlights the use of Taylor series and the CORDIC algorithm for more accurate calculations.

PREREQUISITES
  • Understanding of basic trigonometric functions and their properties
  • Familiarity with the Pythagorean theorem
  • Knowledge of Taylor series for function approximation
  • Basic programming skills for CNC applications
NEXT STEPS
  • Research Taylor series for sine and cosine approximations
  • Learn about the CORDIC algorithm for trigonometric calculations
  • Explore CNC programming techniques for geometric calculations
  • Investigate alternative methods for angle approximation in programming
USEFUL FOR

CNC programmers, mechanical engineers, and anyone involved in numerical control systems who require efficient methods for calculating trigonometric values without built-in functions.

Fergus
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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
 
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You probably just need x^2 + y^2 = r^2
 
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.
 
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.
 
Last edited:
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
 
When evaluating the polar form of complex numbers I hate dealing with angles in the 2nd, 3rd and 4th quadrants.
 
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
 

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