d/dx ( sin x ) = cos x

We know Sin 2x - Sin 2y = 2 cos(x+y) sin(x-y)  (The Trig Equiv page has a proof of this)

So it follows that sin(x+h) - sin(x) = 2 cos(x+h/2) sin(h/2)

If you divide both sides by h, and take the limit as h approaches zero, you get

d/dx (sin x) = cos(x) lim_h−>0 sin(h/2)/(h/2)

That limit is 1, of course (proof) so the result follows.

Related pages in this website

Trigonometric Equivalences

Proof that lim(x−>0) sin x / x = 1

Sin or Cos 3x, 4x, etc. -- trig functions of any multiple of an angle.

The webmaster and author of this Math Help site is Graeme McRae.