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Evaluate (1 + i)^12

First convert to mod-arg form in order to use de Moivre's theorem. |1 + i| = (1^2 + 1^2)^1/2 = 2^1/2 arg(z) is the angle made by the vector of the complex number and the positive real axis. I world recommend...
CL
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Use the geometric series e^(ix) - (1/2)e^(3ix) + (1/4)e^(5ix) - ... to find the exact value sin1 -(1/2)sin3 + (1/4)sin5 - ...

S = e ix - (1/2)e 3ix + (1/4)e 5ix - … is an infinite geometric series, equal to a/(1 - r). a = e ix and r = (1/2)e 2ix thus S = e ix /(1+(1/2)e 2ix = 2e ix /(2+e 2ix ). Rationalising the denominator: S = 2e...
AB
3353 Views

find the sum of r from 0 to n of : 1/((r+1)(r+2)(r+3))

The solution like almost every Methods of Differences questions first involves putting the fraction into partial sums.At this point you would get 3 fractions which can be tricky to deal with. Following what ...
6586 Views

Using z=cos(θ)+isin(θ), find expressions for z^n-1/z^n and z^n+1/z^n

We make use of De Moivre's Theorem which states that (cos( θ)+isin(θ))^n=cos(nθ)+isin(nθ). z^n-1/z^n=cos(nθ)+ isin(nθ)-cos(-nθ)- isin(-nθ)=cos(nθ)+ isin(nθ)-cos(nθ)+ isin(nθ) (by trig relationships)=2isin(nθ...
BS
6561 Views

Show that the points on an Argand diagram that represent the roots of ((z+1)/z)^6 = 1 lie on a straight line.

We want to simplify this equation to one that we know how to solve. If we let ((z+1)/z) = w, then we need to solve w^6 = 1, which is more familiar. Now we try to find the modulus and argument of w. w = re^(i...
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