Complex numbers (i = Sqrt(-1))

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So, a question in my maths book (5th year) reads like so:

Show that (cosΘ + i sinΘ)^2 = cos2Θ + isin2Θ

I cannot figure it out, my problem is with the sin and cos parts as I can handle all the other questions in that excersize pefectly. So can anyone tell me why this is so and give me a quick run down on sin/cos in questions like this. Thanks a lot, Bob.

ecksor

Originally posted by OrangeRhino
So, a question in my maths book (5th year) reads like so:

Show that (cosΘ + i sinΘ)^2 = cos2Θ + isin2Θ

Just multiply it out:

(cosΘ + i sinΘ)^2 = cos^2Θ + 2icosΘsinΘ - sin^2Θ

Then have a look in your log tables, it should have a couple of identities that look like the following two:

cos(2u) = cos^2(u) - sin^2(u)
sin(2u) = 2sin(u)cos(u)

So, cos^2Θ - sin^2Θ becomes cos2Θ and 2i cosΘsinΘ becomes i sin2Θ

M@lice

De Moiver's Theorem! Its also in the log books believe it or not but its sumthin of the form exp(i*Pi).