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### Radian vs. Degree, Sin Derivatives

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

Dr. Math,

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/
```
Associated Topics:
High School Calculus

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