i) Find the exponential and polar form of z_1=-10-6i and its principal value of its argument. ii)...

Question:

i) Find the exponential and polar form of {eq}z_1=-10-6i {/eq} and its principal value of its argument.

ii) For complex number {eq}z_2=4\angle 30^{\circ} {/eq} in polar form, express it in exponential form and then express it in {eq}x+iy {/eq} standard form,

iii) Find {eq}z_1+\bar{z_2} ,\bar{z_2^{2}}, {/eq} and {eq}Im\left ( z_1/z_2 \right ) {/eq}.

Polar and Exponential Form of a Complex Number

{eq}{/eq}

Consider a complex number of the form :

$$z = x + y \ i \\ $$

1. Its modulus (distance from the origin) is given by :

$$r = \sqrt{x^2 + y^2 } \\ $$

2. Its argument is given by the formula :

$$\tan{(arg(z))} = \frac{y}{x} \\ $$

Where, arg(z) is expressed in radians. We need to pick a suitable value of arg(z) such that :

$$-\pi \leq arg(z) \ \leq \pi \\ $$

3. Its polar form is given by

$$z = r\angle \theta \\ $$

{eq}\theta = arg(z) \\ {/eq}

4. Its exponential form is given by :

$$z = r e^{i\theta} \\ $$

{eq}\theta = arg(z) \\ {/eq}

Answer and Explanation:

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{eq}{/eq}

(i) Given complex number :

{eq}z_1 = -10-6i \\ \Rightarrow |z_1| = \sqrt{(-10)^2 + (-6)^2} \\ \Rightarrow |z_1| = \sqrt{136}...

See full answer below.


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Complex Numbers in Polar Form: Process & Examples

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Chapter 24 / Lesson 2
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After watching this video lesson, you will be able to convert complex numbers from rectangular form to polar form easily by following the formulas you will see here. You will also learn how to find the power of a complex number.


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