Question

Given the complex number z_1=3\big(\cos \frac{14\pi}{15} +i\sin \frac{14\pi}{15}\big)z 1 ​ =3(cos 15 14π ​ +isin 15 14π ​ ) and z_2=3\sqrt{3}\big(\cos \frac{11\pi}{15} +i\sin \frac{11\pi}{15}\big)z 2 ​ =3 3 ​ (cos 15 11π ​ +isin 15 11π ​ ), express the result of z_1z_2z 1 ​ z 2 ​ in rectangular form with fully simplified fractions and radicals.

Answer 1

The product of z₁ = 3 · (cos 14π/15 + i · sin 14π/15) and z₂ = 3 √3 · (cos 11π/15 + i · sin 11π/15) in rectangular form with fully simplified expressions is z₁ · z₂ = 7.794 - i · 13.5.

How to determine the product of two complex numbers

Let be two numbers of the form z = a + i · b, where i = √-1, the product of two of these numbers in rectangular form is described by the following formula:

z₁ · z₂ = (a + i · b) · (c + i · d) = (a · c - b · d) + i · (a · d + b · c)   (1)

If we know that a = 3 · cos 14π/15, b = 3 · sin 14π/15, c = 3√3 · cos 11π/15, d = 3√3 · sin 11π/15, then the result in rectangular form is:

z₁ · z₂ = 7.794 - i · 13.5

The product of z₁ = 3 · (cos 14π/15 + i · sin 14π/15) and z₂ = 3 √3 · (cos 11π/15 + i · sin 11π/15) in rectangular form with fully simplified expressions is z₁ · z₂ = 7.794 - i · 13.5. $\blacksquare$

Remark

The statement presents typing mistakes and is poorly formatted, the correct form is introduced below:

Given the complex number z₁ = 3 · (cos 14π/15 + i · sin 14π/15) and z₂ = 3 √3 · (cos 11π/15 + i · sin 11π/15), express the result of z₁ · z₂ in rectangular form with fully simplified fractions and radicals.

To learn more on complex numbers, we kindly invite to check this verified question: brainly.com/question/10251853