%5E5-in-polar-form.jpeg "(1+i)^5 In Polar Form")
Finding (1 + i)^5 in Polar Form
This article explores how to calculate (1 + i)^5 in polar form.
Understanding Complex Numbers in Polar Form
A complex number, z, can be represented in rectangular form as z = a + bi, where a and b are real numbers and i is the imaginary unit (i^2 = -1). It can also be expressed in polar form as:
z = r(cos θ + i sin θ)
Where:
- r is the magnitude or modulus of z, calculated as √(a^2 + b^2).
- θ is the argument or angle, measured counter-clockwise from the positive real axis, and calculated as tan⁻¹(b/a).
Converting (1 + i) to Polar Form
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Magnitude (r): r = √(1² + 1²) = √2
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Argument (θ): θ = tan⁻¹(1/1) = 45° or π/4 radians
Therefore, (1 + i) in polar form is √2(cos π/4 + i sin π/4).
De Moivre's Theorem
De Moivre's Theorem provides a straightforward way to calculate powers of complex numbers in polar form. It states:
(cos θ + i sin θ)^n = cos(nθ) + i sin(nθ)
Calculating (1 + i)^5
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Apply De Moivre's Theorem: (√2(cos π/4 + i sin π/4))^5 = (√2)^5 (cos (5π/4) + i sin (5π/4))
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Simplify: = 4√2 (cos (5π/4) + i sin (5π/4))
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Evaluate the trigonometric functions: = 4√2 (-√2/2 - i√2/2)
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Simplify further: = -4 - 4i
Therefore, (1 + i)^5 in polar form is 4√2 (cos (5π/4) + i sin (5π/4)), which simplifies to -4 - 4i in rectangular form.