[RESOLVED] VB.NET - Attempting to write a custom sine function

Hi,

Curious to learn a bit more about trigonometry and trying to get a better understanding of the underlying functions (sine, cosine, tangent, etc.) I tried to write a custom sine function in VB.NET:

Code:

---

Imports System

Module Module1
  Sub Main()
      Console.WriteLine(Math.Sin(Math.PI / 2))
      Console.WriteLine(CustomSine(Math.PI / 2))
  End Sub

  Function CustomSine(Angle As Double) As Double
      Return Angle - (Angle ^ 3.0 / 6.0) + (Angle ^ 5.0 / 120.0)
  End Function
End Module

---

The "CustomSine" function above is based on information found at https://stackoverflow.com/questions/...-code-for-sinx. Specifically the following algebraic formula:

Code:

---

x - (x^3/3!) + (x^5/5!)

---

As I understand it "3!" and "5!" simply mean "give me the factorials for 3 and 5". Seeing no reason to write a custom function (did that once already and would only slow the "CustomSine" function down) for the factorials of two numeric constants the precalculated values (6.0 and 120.0 respectively) were used instead.

As to the question:
My "CustomSine" produces consistently higher values than the "System.Math.Sin" function. As I understand the output values should never exceed the range of -1 to +1 when using 0 to PI / 2.0 as inputs. Either my code has errors (likely rounding errors) or the approximation given at StackOverflow isn't accurate enough. What formula would produce better results and why?

Just to be clear, this is just for a hobby and higher math isn't my specialty. Thanks.

PI / 2 is approx 1.570796327 (call it x) and sine(x) should give an answer of 1

(x ^ 3) / 6 = 0.645964098

(x ^ 5) / 120 = 0.0796926262

So sine(x) is 1.570796327 - 0.645964098 + 0.0796926262 = 1.0045248552

which is correct 2 dp. If you need more accuracy then you can consider further terms in the series.

The next two terms are -(x ^ 7) / 7! and (x ^ 9) / 9! (and then -(x ^ 11) / 11! and (X^13) / 13! )

Quote:

---

Originally Posted by 2kaud

PI / 2 is approx 1.570796327 (call it x) and sine(x) should give an answer of 1

(x ^ 3) / 6 = 0.645964098

(x ^ 5) / 120 = 0.0796926262

So sine(x) is 1.570796327 - 0.645964098 + 0.0796926262 = 1.0045248552

which is correct 2 dp. If you need more accuracy then you can consider further terms in the series.

The next two terms are -(x ^ 7) / 7! and (x ^ 9) / 9! (and then -(x ^ 11) / 11! and (X^13) / 13! )

---

Fixed! Thanks.

Code:

---

  Function CustomSine(Angle As Double) As Double
      Return Angle - (Angle ^ 3.0 / 6.0) + (Angle ^ 5.0 / 120.0) - (Angle ^ 7.0 / 5040.0) + (Angle ^ 9.0 / 362880) - (Angle ^ 11.0 / 39916800.0) + (Angle ^ 13.0 / 6227020800.0) - (Angle ^ 15.0 / 1307674368000.0) + (Angle ^ 17.0 / 355687428096000.0) - (Angle ^ 19.0 / 121645100408832000)
  End Function

---

Any factorials beyond 19 appear to be pointless and result in an overflow anyway.

Great - but you usually won't implement like that.

Note that if t is the term number (starting at 1)

third term (t = 3)
Angle ^ 5 = <prev value> * Angle * Angle
5! = <prev value> * 4 * 5
note that 4 is (t - 1) * 2 and 5 is (t - 1) * 2 + 1

fourth term (t = 4) where <prev value> is that obtained from the third term
Angle ^7 = <prev value> * Angle * Angle
7! = <prev value> * 6 * 7
note that 6 is (t - 1) * 2 and 7 is (t - 1) * 2 + 1

etc etc

and that the signs of the terms alternate between + and -.

Given this, the calculation can be done in a simple loop to the required number of terms.

This code has been removed.