1. ## Complex Numbers

In order to rotate a molecule through an angle (theta), a molecular graphics package needs to be able to calculate the new coordinates (x-prime, y-prime) of every point in terms of its previous coordinates (x, y) and the angle (theta).

i. Use the fact that, in the complex plane, multiplication by e^i(theta) produces a rotation through an angle (theta) to derive expressions for x-prime and y-prime in terms of x, y and the angle (theta)

ii. By expressing x and y in terms of polar coordinates, demonstrate that the following identities hold for all angles A and B:
a) sin (A+B) = sinA cosB + cosA sinB
b) cos (A+B) = cosA cosB - sinA sinB

2. Originally Posted by superdael
In order to rotate a molecule through an angle (theta), a molecular graphics package needs to be able to calculate the new coordinates (x-prime, y-prime) of every point in terms of its previous coordinates (x, y) and the angle (theta).

i. Use the fact that, in the complex plane, multiplication by e^i(theta) produces a rotation through an angle (theta) to derive expressions for x-prime and y-prime in terms of x, y and the angle (theta)
I denote x-prime a nd y-prime by a and b respectively.
We have
$a+ib=(x+iy)e^{i/theta}$
where $e^{i/theta}=cos/theta +isin/theta$
compare real and imaginary parts, you will get it.
ii. By expressing x and y in terms of polar coordinates, demonstrate that the following identities hold for all angles A and B:
a) sin (A+B) = sinA cosB + cosA sinB
b) cos (A+B) = cosA cosB - sinA sinB
$e^{iA}=cosA +isinA$
and $e^{iB}=cosB +isinB$
multiply the above two equation and compare real and imaginary parts

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