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Definition of Cosine
By Niranjan Srinivas on Wednesday, February 27, 2002 - 02:25 am:

In the result eix = cosx + isinx.....(1)
what is the definition of cosx ?

If cosx is defined as the series obtained,then the proof for (1) will become a matter of definition. If cosx is defined as the ratio adj/hyp in any right angled triangle, then how do we show that that series and this ratio are the same ?
can anybody enlighten me on this ?
Thanks !
Cheers,
Niranjan

By Tristan Marshall on Thursday, February 28, 2002 - 03:12 pm:

You were talking about the ratios for cos and sin:

cos = adj/hyp sin = opp/hyp

This is probably the way you first met sin and cos. From a few more trig identities, and a bit of geometry, we can 'prove' the derivatives of cos & sin:

d/dx(sin x) = cos x

d/dx(cos x) = -sin x

Then we can use Maclaurin's theorem. This says that if a function f is infinitely differentiable at 0, then

f(x) = f(0) + f(1)(0)x + [f(2)(0)/2!]x2 + ... + [f(r)(0)/r!]xr + ...

where this series converges. It so happens that for cos and sin we get series that converge everywhere, and these series are:

cos x = 1 - x2/2! + x4/4! + ... + (-1)rx2r/(2r)! + ...

sin x = x - x3/3! + ... + (-1)r+1x2r+1/(2r+1)! + ...

Also ex = 1 + x2/2! + x3/3! + ...

And the formula for eiq can be obtained by comparing these series.

However, in higher analytical work, it is more usual to start by defining sin and cos as the power series, and then deducing the ratios. We do this using the formula for eiq.

The set of points z = eiq 0 <= q < 2p form a unit circle in the complex plane.

For this next bit, a diagram would help, but I can't manage to draw a good one, so you might want to try sketching one yourself.

If we choose a point z in the first quadrant of this circle (ie real and imaginary parts >0) given by z = eiq and draw a line connecting z to the origin, then the formula for eiq tells us that the angle between this line and the positive real axis is q

To see this, drop a line down from z to the real axis such that this line is perpendicular to the real axis. You now have a right-angled triangle, where the hypotenuse has length 1, the adjacent side has length cosq (the real part of z) and the opposite side has length sinq (the imaginary part of z).

This shows that the ratios work when the hypotenuse has lenght 1. By scaling, we see it is true for all right-angled triangles.