/*! 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 6 Write \(-3+3 \sqrt{3} i\) in pol... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 6

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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Most popular questions from this chapter

Suppose \(w\) and \(z\) are complex numbers such that the real part of \(w z\) equals the real part of \(w\) times the real part of \(z\). Explain why either \(w\) or \(z\) must be a real number.

Write each expression in the form \(a+b i,\) where \(a\) and \(b\) are real numbers. $$ (5+7 i)+(4+6 i) $$

Show that if \(p\) is a polynomial with real coefficients, then $$ p(\bar{z})=\overline{p(z)} $$ for every complex number \(z\).

Verify that $$ (\sqrt{3}+i)^{6}=-64 $$.

Show that multiplication of complex numbers is commutative, meaning that $$ w z=z w $$ for all complex numbers \(w\) and \(z\).
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