d (cos x + dx i sin x) = sin x + i cos x = i(cos x + i sin x) so cos x + i sin x has the correct derivative to be the desired extension of the exponential function to the case c = i.

This is very useful information about the function sin(x) but it doesn’t tell the whole story. For example, it’s hard to tell from the formula that sin(x) is periodic.

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1 Trigonometric Identities you must remember The “big three” trigonometric identities are sin2 t + cos2 t = 1 sin(A + B) = sin A cos B + cos A sin B we can derive many other identities. Even if we commit the …

Note that you can get (5) from (4) by replacing B with -B, and using the fact that cos(-B) = cos B (cos is even) and sin(-B) = - sin B (sin is odd). Similarly (7) comes from (6).

TRIGONOMETRIC FUNCTIONS WITH eax (95) ex sin xdx = ! 1 ex [ sin x " cosx ]

Theorem. sin z = 0 () z = n1⁄4 for some integer n. Proof. By trigonometry we know that sin 1⁄4n = 0 for any integer n, so what's at stake here is the converse: if sin z = 0 then z = 1⁄4n for some integer n.