Question

convert the complex number -7-24i to its polar form

Answer 1

Answer:

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Step-by-step explanation:

$so \: given \: complex \: number \: is \\ z = - 7 - 24i \\ \\ comparing \: it \: with \: \: a + ib \: \\ we \: have \\ \\ a = - 7 \\ b = - 24 \\ \\ so \: we \: know \: that \\ \\ |z| = r = \sqrt{a {}^{2} + b {}^{2} } \\ \\ = \sqrt{( - 7) {}^{2} + ( - 24) {}^{2} } \\ \\ = \sqrt{49 + 576} \\ \\ = \sqrt{625} \\ \\r = 25$

$similarly \: then \\ \\ arg \: (z) = θ = tan {}^{ - 1} ( \: \frac{b}{a} \: ) \\ \\ = tan {}^{ - 1} ( \frac{ - 24}{ - 7} ) \\ \\ θ= tan {}^{ - 1} \: \frac{24}{7} \\ \\ we \: have \\ \\ polar \: form \: of \: z = r.e {}^{i \:θ } \\ \\ = 25.r {}^{ \: tan {}^{ - 1} \frac{ \: 24}{7} i}$

Answer 2

Given :-

Complex number is $-7-24i$

To find :-

Polar form of the given complex number

Solution :-

Inorder to represent any complex number in it's polar form, firstly we need to calculate the modulus and argument of the complex number.

Let's assume that,

• $y = Im(z)$
• $x = Re(z)$

• Now, the modulus of the complex number is given by,

$\small {\implies|z| = \sqrt{Re(z)^{2} +Im(z)^{2}} }$

$\small{\implies|z| = \sqrt{ x^{2} + {y}^{2}} }$

$\small{\implies|z| = \sqrt{ ( - 7)^{2} + { ( - 24)}^{2}} }$

$\small {\implies|z| = \sqrt{ 49 + 576} }$

$\small {\implies|z| = \sqrt{625} }$

$\small\boxed {\implies|z| = 25}$

• Now the argument of the complex number in 3rd quadrant is given by,

$\small \implies Arg(z) = \tan^{-}\bigg|\dfrac y x\bigg| -\pi$

$\small {\implies Arg(z) = \tan^{-}\bigg|\dfrac { - 24}{ - 7}\bigg| -\pi}$

$\small{\implies Arg(z) = \tan^{-}\bigg|\dfrac { 24}{ 7}\bigg| -\pi}$

$\small \boxed {\implies Arg(z) = \tan^{-}\bigg(\dfrac { 24}{ 7}\bigg)-\pi}$

• Now, the polar form of complex number is given by,

$\small \longrightarrow r \Big( \cos\theta + i \sin\theta\Big)$

Here,

• $r = |z|$
• $\theta = Arg(z)$

Therefore, by substituting the values, we get :

$\small{ \longrightarrow \underline{\underline{25\Big( \cos {\left[\tan^{-}\left(\dfrac { 24}{ 7}\right)-\pi\right] + i \sin\theta\left[\tan^{-}\left(\dfrac { 24}{ 7}\right)-\pi\right]\Big)}}}}$

This is the required polar form of the given complex number.

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Note :-

This can also be expressed in short as,

$\small \underline{ \boxed{\rm 25 \: cis \left[\tan^{-}\left(\dfrac { 24}{ 7}\right)-\pi\right]}}$

Here, cis stands for cos, iota and sin.

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Learn More :

Express the complex number √2+4i in the polar form.

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