De Moivres Theorem Calculator

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Example

Created on 2024-06-20Asked by Amelia Scott (Solvelet student)

Use De Moivre's theorem to find

(1 + i)^4

Solution

Step 1: Identify the complex number

z = 1 + i

and the exponent

n = 4

. Step 2: Apply De Moivre's theorem:

z^n = \left(r \operatorname{cis} \theta\right)^n = r^n \operatorname{cis}(n\theta),

where

r = |z|

and

\theta = \operatorname{arg}(z)

. Step 3: Calculate

r

and

\theta

r = |1 + i| = \sqrt{1^2 + 1^2} = \sqrt{2},

\theta = \operatorname{arg}(1 + i) = \arctan(1).

Step 4: Substitute the values into De Moivre's theorem:

(1 + i)^4 = (\sqrt{2})^4 \operatorname{cis}(4\arctan(1)).

Step 5: Calculate

(\sqrt{2})^4

and

4\arctan(1)

(\sqrt{2})^4 = 2^2 = 4,

4\arctan(1) = 4(\frac{\pi}{4}) = \pi.

Step 6: Conclusion.

(1 + i)^4 = 4 \operatorname{cis}(\pi) = -4

. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Luna Lee on Solvelet

1. Use De Moivre's theorem to find the fifth roots of the complex number

z = 1 + i

2. Compute the 10th power of the complex number

w = \cos\left(\frac{\pi}{4}\right) + i \sin\left(\frac{\pi}{4}\right)

DefinitionDe Moivres theorem is simply the result that allows the fixed-point theorem to be valid, where for any complex number z, which is equal to r(cosθ+isinθ) (where r is the magnitude and θ the angle) and integer n, we have zk = (r(cos(θ)+isin(θ)))k = r(cos(kθ)+isin(kθ)). It plays a key role for trigonometric identities, and solving of (for example) differential equations in complex analysis as well. Eg : De Moivre's theorem can be applied to determine the roots of complex numbers, i.e. the square roots or cube roots of complex numbers.

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