Question about numerical integration for cos vs cosd

Hi all,

I came across this question yesterday. Here is the code:

syms theta
y = double(int(cos(theta),theta,[0 pi/12]));

The answer, of course, y = 0.2588.

However, when I use cosd instead:

y = double(int(cosd(theta),theta,[0 15]));

Then y = 14.8292.

If I do it by hand the answer is y = 0.2588 for both case.

I'm wondering what happened when we use cosd with int.

I also checked WolframAlpha and it gives me those two different answers as well. Did I misunderstandd anything during my academic years?

Answer 1

[James Tursa]

By hand:

int cosd(theta)

= int cos(theta*pi/180)

= sin(theta*pi/180)*180/pi

Integrated from 0 to 15 degrees:

sin(15*pi/180)*180/pi - sin(0*pi/180)*180/pi = 14.8292

Looks like MATLAB got the correct answer to me.

The key is that derivatives and integrals of cosd (and sind etc.) have to account for the rad/deg factor.

I.e., d/dtheta sind(theta) ≠ cosd(theta)

Rather, d/dtheta sind(theta) = cosd(theta) * (pi/180)

To see why this must be so, plot up both cos(x) from 0 to pi/12 and cosd(x) from 0 to 15 on the same plot. Which one has the largest area underneath it? The cosd(x) plot, of course, since it has a lot more x length under it.

[Chi So]

Thanks James!