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

### Sin(x) = x?

```
Date: 07/01/99 at 08:14:20
From: Mohamed YOUNES
Subject: Sin(x) = x?

Dear Dr.Math,

One of my Maths teachers told me that when x is very small (for
example, x = 10^-70) then sin(x) = x. Is that true? Why?

Thank you.
```

```
Date: 07/01/99 at 12:56:00
From: Doctor Peterson
Subject: Re: Sin(x) = x?

Hi, Mohamed.

The only value for which sin(x) = x is 0. However, the line y = x is
tangent to y = sin(x) at that point, so for very small values of x
sin(x) will be very close to x:

|         /     *******
|       / ***
|     /**
|   **
| *
-------------------- *-----------------------
* |
**   |
**/     |
*** /       |
******    /         |

Incidentally, this is one reason why it is natural to use radians to
define the trigonometric functions: it makes the slope of the sine
(and tangent) at zero be 1.

You can see why this is true if you consider the definition of the
sine as the y coordinate of the point on the circle at a given angle:

***********
****     |     ****
****         |         ****
*             |           /|*
**              |         /  | **
*                |       /    |   * A
*                 |     /      |sin(A)
*                 |   /        |    *
*                  | /A         |     *
*------------------*------------+-----*
-1*                  |                  *1
*                 |                 *
*                 |                 *
*                |                *
**              |              **
*             |             *
****         |         ****
****     |     ****
***********

For very small angles, the sine of A is very close to the length of
the arc, which is A (again because of the way we've defined the
radian), because the circle becomes very nearly vertical.

questions from today, but I hope you'll keep asking one question at a
time until you know all there is to know.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Trigonometry

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