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/ |
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