Radian vs. Degree, Sin Derivatives

Date: 10/23/2001 at 23:51:41
From: Liz
Subject: Radian and degree, sin derivatives

Dr. Math,

Why do you get two different answers when you ask your calculator for 
the derivative of sinx in degree and radian modes? When you have it in 
radian mode, it gives you the right answer (the cosx). When you put it 
in degree mode, it gives you pi*cosx/180. Why is that?

Liz

Date: 10/24/2001 at 17:02:57
From: Doctor Peterson
Subject: Re: Radian and degree, sin derivatives

Hi, Liz.

In degree mode, the calculator is essentially replacing the sine 
function with a different function whose argument is a number of 
degrees. If we call this new function sind, we can define it as

    sind(x) = sin(pi/180 x)

since to calculate it we have to convert x degrees to radians and then 
take the actual sine.

Take the derivative of this with respect to x, and you get

    d/dx sind(x) = pi/180 cos(pi/180 x)

                 = pi/180 cosd(x)

Of course, what the calculator calls the cosine is now really this 
"cosd" function.

Does that help? This also explains why we use radians when we do 
calculus. Like many things mathematicians do, this makes everything 
easier.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/