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.

I'm glad you enjoy math as I do. You won't get answers to all your 
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/