Finding Coordinates of Partial Arc Radius Without Sine and Cosine Functions

  • Thread starter Fergus
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In summary, the conversation was about finding x and y coordinates of a partial arc radius given the angle for a program on a CNC lathe. The issue was that the controller did not have sine and cosine functionality and the individual needed help recalling a basic definition from trigonometry. A possible solution was using the Pythagorean theorem or approximating the sine and cosine functions. Eventually, the use of Taylor series and CORDIC algorithm was suggested and the individual found the solution in an old math book.
  • #1
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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  • #2
You probably just need x^2 + y^2 = r^2
 
  • #3
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.
 
  • #4
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.
 
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  • #5
Yep wat uart said was 100% correct, and for the cosine function you may want to use [tex]cos(x)[/tex] = [tex]\pi[/tex] - [tex](x*\pi/180)^{2}/2[/tex] with similar accuracy. for more accuracy just add ...[tex]+ (x*\pi/180)^{4}/24[/tex]
 
  • #6
When evaluating the polar form of complex numbers I hate dealing with angles in the 2nd, 3rd and 4th quadrants.
 
  • #7
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
 

1. What is the definition of sine, cosine, and tangent?

Sine, cosine, and tangent are trigonometric functions that represent the ratios of the sides of a right triangle. Sine (sin) is the ratio of the opposite side to the hypotenuse, cosine (cos) is the ratio of the adjacent side to the hypotenuse, and tangent (tan) is the ratio of the opposite side to the adjacent side.

2. How do you find the values of sine, cosine, and tangent on a unit circle?

To find the values of sine, cosine, and tangent on a unit circle, you can use the acronym SOH-CAH-TOA. Sine is the y-coordinate, cosine is the x-coordinate, and tangent is the y-coordinate divided by the x-coordinate. You can also use a calculator to find these values.

3. What is the Pythagorean identity?

The Pythagorean identity is a fundamental trigonometric identity that states: sin^2(x) + cos^2(x) = 1. This identity is derived from the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

4. How do you solve for missing sides and angles in a right triangle?

To solve for missing sides and angles in a right triangle, you can use the trigonometric ratios (sine, cosine, tangent) and the Pythagorean theorem. If you know the values of two sides or one side and one angle, you can use the appropriate trigonometric ratio to find the missing side or angle. If you know the values of two sides, you can use the Pythagorean theorem to find the length of the third side.

5. What are the common trigonometric values for special angles?

The common trigonometric values for special angles (in degrees) are as follows: sin(0) = 0, cos(0) = 1, tan(0) = 0, sin(30) = 1/2, cos(30) = sqrt(3)/2, tan(30) = 1/sqrt(3), sin(45) = sqrt(2)/2, cos(45) = sqrt(2)/2, tan(45) = 1, sin(60) = sqrt(3)/2, cos(60) = 1/2, tan(60) = sqrt(3), sin(90) = 1, cos(90) = 0, tan(90) = undefined. These values can be found on the unit circle or by using a calculator.

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