Question

Please answer that question

Answer 1

The polar form of the complex numbers are z₁ = 6 · $e^{i\,\frac{11\pi}{6} }$ and z₂ = √2 · $e^{i\,\frac{3\pi}{4} }$. The multiplication of the numbers is z₁ · z₂ = 6√2 · $e^{i\,\frac{31\pi}{12} }$, the division of the numbers is z₁ / z₂ = 3√2 · $e^{i\,\frac{13\pi}{12} }$ and the reciprocal of a number is 1 / z₁ = (1 / 6) · $e^{-i\,\frac{11\pi}{6} }$.

How to find complex numbers in polar forms and make operations with them

In this problem we find two complex numbers in rectangular form (z = a + i b), whose polar form has to be found. The complex form is described below:

z = r · $e^{i\,\theta}$

Where:

• r - Norm
• θ - Direction, in radians.

The magnitude is determined by Pythagorean theorem:

r = √(a² + b²)

And the direction by inverse trigonometric functions:

θ = tan⁻¹ (b / a)

Polar form offers a quicker manner to perform multiplication and division of complex numbers. Thus:

Multiplication

z₁ · z₂ = r₁ · r₂ · $e^{i\,(\theta_1 + \theta_2)}$

Division

z₁ / z₂ = (r₁ / r₂) · $e^{i\,\frac{\theta_{1}}{\theta_{2}} }$

First, determine the magnitudes of each complex number:

r₁ = √[(3√3)² + (- 3)²]

r₁ = 6

r₂ = √[(- 1)² + 1²]

r₂ = √2

Second, determine the directions of each complex number:

θ₁ = 11π / 6

θ₂ = 3π / 4

Third, write the complex numbers in polar form:

z₁ = 6 · $e^{i\,\frac{11\pi}{6} }$

z₂ = √2 · $e^{i\,\frac{3\pi}{4} }$

Fourth, find the multiplication of the complex numbers:

z₁ · z₂ = 6√2 · $e^{i\,\frac{31\pi}{12} }$

Fifth, find the division of the complex numbers:

z₁ / z₂ = 3√2 · $e^{i\,\frac{13\pi}{12} }$

Sixth, find the reciprocal of the complex number:

1 / z₁ = (1 / 6) · $e^{-i\,\frac{11\pi}{6} }$

To learn more on complex numbers: brainly.com/question/20566728

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