Math related question

Discussion around the Swiss Micros DM42 calculator.
mcc
Posts: 210
Joined: Fri Jun 23, 2017 3:10 am

Math related question

Hi,

If (for example) I integrate sine(x) from 0...180 degree with my DM42 I get: 114.591577478...
If I do the same thing in RAD mode from 0...Pi -- again with my DM42 -- I get: 2.00000032205...

If I do an RAD->DEG conversion, I get the first result back.
If I divide the first example by 180 and the second by Pi I get both times: 0.63661977237.

But...WHAT is this value...for "what it is te value for" ? It looks like an integral value which represents
the ??? independantly from the "base" the integration process relies on....

Is it something known in mathematics...or is it simply nonsense?

Thanks a lot for any information on this in advance!
Cheers!
mcc
DM 42 - SN: 00373, Firmware v.:3.14. / 3.14. as compiled by SwissMicros

Thomas Okken
Posts: 591
Joined: Tue May 02, 2017 3:48 pm
Location: New Jersey, USA
Contact:

Re: Math related question

That value is 2/pi, since the integral from 0 to pi of sin(x), in radians, is exactly 2. Your DM42 will give a more accurate result if you set ACC to something smaller.

The integral of sin(x), in degrees, from 0 to 180, is exactly 360/pi, since you're basically stretching one half-wave by a factor of 180/pi. Or, more formally, sin_deg(x) = sin(x*pi/180), so

integ(0, 180, sin_deg(x) dx)
= integ(0, 180, sin(x*pi/180) dx)
[substituting y = x*pi/180]
= integ(0, pi, sin(y) dx)
[given that dy/dx = pi/180 so dx = dy*180/pi]
= integ(0, pi, (sin(y)*180/pi) dy)
[taking the constant out of the integral]
= 180/pi*integ(0, pi, sin(y) dy)

Proving that integ(0, pi, sin(x) dx) = 2 is left as an exercise to the reader. You are allowed to assume that the antiderivative of sin(x) is -cos(x).