%5E3-in-polar-form.jpeg "(1+i)^3 In Polar Form")
Finding the Polar Form of (1 + i)³
In this article, we will explore how to express the complex number (1 + i)³ in polar form.
Understanding Polar Form
The polar form of a complex number represents it in terms of its magnitude (or modulus) and angle (or argument). It is expressed as:
z = r(cos θ + i sin θ)
Where:
- r is the magnitude of the complex number, calculated as √(a² + b²) for a complex number (a + bi).
- θ is the angle the complex number makes with the positive real axis, measured in radians, and calculated as tan⁻¹(b/a).
Finding the Polar Form of (1 + i)
1. Calculate the magnitude (r):
For (1 + i), a = 1 and b = 1.
Therefore, r = √(1² + 1²) = √2.
2. Calculate the angle (θ):
θ = tan⁻¹(b/a) = tan⁻¹(1/1) = π/4 radians.
3. Polar Form of (1 + i):
The polar form of (1 + i) is √2(cos(π/4) + i sin(π/4)).
Finding the Polar Form of (1 + i)³
We can use De Moivre's theorem to simplify the process:
(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)
Applying this to (1 + i)³, we get:
(1 + i)³ = [√2(cos(π/4) + i sin(π/4))]³ (1 + i)³ = (√2)³ (cos(3π/4) + i sin(3π/4)) (1 + i)³ = 2√2 (cos(3π/4) + i sin(3π/4))
Conclusion
Therefore, the polar form of (1 + i)³ is 2√2(cos(3π/4) + i sin(3π/4)). This representation provides a clear understanding of the complex number's magnitude and direction in the complex plane.