The complex number in rectangular form z = (5√3 / 4) - i 5/4 is equivalent to the complex number in polar form z = 5/2 · (cos 11π/6 + i sin 11π/6). (Correct choice: C)
How to transform a complex number in rectangular form into polar form
Complex numbers are elements of the form z = a + i b, where [tex]a, b \in \mathbb{R}[/tex]. In other words, represents a generalization from real numbers. The polar form of a complex number is shown below:
[tex]z = r\cdot (\cos \theta + i \,\sin \theta)[/tex] (1)
Where:
- r - Norm
- θ - Direction, in radians.
The norm is determined by Pythagorean theorem and the direction by inverse trigonometric reason. If we know that z = (5√3 / 4) - i 5/4, then its polar form is shown below:
Norm
[tex]r = \sqrt{\left(\frac{5\sqrt{3}}{4} \right)^{2}+\left(-\frac{5}{4} \right)^{2}}[/tex]
r = 5/2
Direction
[tex]\theta = \tan^{-1} \left[\frac{\left(-\frac{5}{4} \right)}{\left(\frac{5\sqrt{3}}{4} \right)} \right][/tex]
θ = 11π/6 rad
Thus, the complex number in rectangular form z = (5√3 / 4) - i 5/4 is equivalent to the complex number in polar form z = 5/2 · (cos 11π/6 + i sin 11π/6).
To learn more on complex numbers: https://brainly.com/question/10251853
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