/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 16 Write \(-3+3 \sqrt{3} i\) in pol... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 16

Write \(-3+3 \sqrt{3} i\) in polar form.

Short Answer

Expert verified
The polar form of \(-3 + 3\sqrt{3}i\) is \(6(\cos(\arctan(-\sqrt{3}) + \pi) + i\sin(\arctan(-\sqrt{3}) + \pi))\).

Step by step solution

01

Identify the given complex number

In this problem, the given complex number is "-3 + 3鈭�3 i". Let this complex number be represented as z = a + bi, where a = -3 and b = 3鈭�3.
02

Calculate the modulus (r) of the complex number

The modulus of the complex number z = a + bi is given by the formula: r = \(\sqrt{a^2 + b^2}\) So, in this case, a = -3 and b = 3鈭�3. Plug in the values: r = \(\sqrt{(-3)^2 + (3鈭�3)^2}\) r = \(\sqrt{9 + 27}\) r = \(\sqrt{36}\) r = 6 Thus, the modulus of the given complex number is 6.
03

Calculate the argument (胃) of the complex number

The argument (or angle) of a complex number z = a + bi is given by the formula: 胃 = arctan(\(\frac{b}{a}\)) In this case, a = -3 and b = 3鈭�3. Plug in the values: 胃 = arctan(\(\frac{3\sqrt{3}}{-3}\)) 胃 = arctan(-鈭�3) Now, since a < 0 and b > 0, the complex number lies in the second quadrant of the complex plane. Therefore, we need to add 蟺 to the angle. 胃 = arctan(-鈭�3) + 蟺
04

Write complex number in polar form

Using the calculated r and 胃 values, we convert the complex number to polar form. The polar form of a complex number is given by z = r(cos(胃) + i sin(胃)). In this case, r = 6 and 胃 = arctan(-鈭�3) + 蟺. Plug in the values: z = 6(cos(arctan(-鈭�3) + 蟺) + i sin(arctan(-鈭�3) + 蟺)) This is the polar form of the given complex number -3 + 3鈭�3 i.

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