Associated Topics || Dr. Math Home || Search Dr. Math

### Find x in Degrees, not Radians

```
Date: 01/14/98 at 11:26:59
From: Ronny
Subject: Trigonometry/Derivatives

Hi Dr. Math,

I Just have one quick question. We have learned in Math class that the
derivative of sin(x) is cos(x). My teacher said this implies that the
x is in radians, not degrees. My teacher gave us a homework assignment
and one of the problems is to find the derivative of sin(x) when x is
in degrees, not radians. I have no idea how to prove this. Could you

Thank you very very much in advance.

Mathematically Yours,
Ronny
```

```
Date: 02/10/98 at 15:45:01
From: Doctor Sonya
Subject: Re: Trigonometry/Derivatives

The key to this problem is the lim   (sin x)/x
x->0

where they do the proof of that result. The important point is that
if x is in radians, then for values of x close to 0, sin x and x are
approximately the same.

You also need to verify what happens to (1 - cos x)/x as x-> 0 when
x is in degrees. This one will still be 0, just as in radians.

Let's suppose, instead, that x is in degrees. Then sin x is
approximately the same as x * Pi/180. This is because sin x is
approximately equal to x. If we want to convert x from radians to
degrees, we have to multiply x by Pi/180, so sin(x) = x(Pi/180).
Therefore we see that for x in degrees,

sin x      Pi
lim   ------- = -----
x->0     x       180

Now, let's go to derivatives.

d(sin x)       sin(x+h) - sin x      sin x cos h + sin h cos x - sin x
------- = lim  --------------- = lim ---------------------------------
dx      h->0        h           h->0               h

sin h             1 - cos h    Pi
= lim cos x ----- - lim sin x --------- = ---- cos x - sin x (0)
h->0        h     h->0           h       180

Pi
= ---- cos x
180

I hope this helps.

-Doctors Sonya and Fred,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Calculus

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search