How this calculator works
Divide the components by the magnitude to obtain the cosine and sine of the direction. Their squares add to one. The principal phase comes from atan2, which accounts for all quadrants. De Moivre’s theorem raises the magnitude to the chosen integer power and multiplies the angle by that power. The five-row table illustrates this rule. If any result exceeds the finite numeric range, use smaller components.
Formula
r=√(a²+b²); θ=atan2(b,a); z=r(cos θ+i sin θ); zⁿ=rⁿ[cos(nθ)+i sin(nθ)].
Worked example
For z=1+i, r=√2 and θ=45°, so z=√2(cos 45°+i sin 45°). Squaring gives 2i and taking the fourth power gives -4. For z=i the first four powers are i, -1, -i and 1.
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Frequently asked questions
Is trigonometric form different from exponential form?
They express the same number. Euler’s identity makes cos θ+i sin θ equal to exp(iθ); this page keeps the sine and cosine factors visible.
What does the unit direction show?
It removes size and leaves the point on the unit circle in the same direction as the original complex number.
Can zero have a unit direction?
No. Its magnitude is zero, so dividing by magnitude is undefined. The phase and unit components are omitted, while positive integer powers remain zero.