Question

How do you write the complex number $6 - 8i$ in polar form?

Answer 1

Hint: According to the question, we have to write the complex number $6 - 8i$ in polar form that is $z = r(\cos \theta + i\sin \theta )$
So, first of all we have to find $r$ and $\theta $ with the help of the formula mentioned below.

Formula used:
$ \Rightarrow r = \sqrt {{x^2} + {y^2}} ....................(A)$
$ \Rightarrow \theta = {\tan ^{ - 1}}\left( {\dfrac{y}{x}} \right)............................(B)$
where $x$ and $y$ are the real and imaginary parts respectively of the given complex number $6 - 8i$ that is in the form of $z = x + iy$.

Complete step-by-step answer:
Step 1: First of all we have to let that $z = 6 - 8i$
Now, we have to compare this equation $z = 6 - 8i$to the standard form of the complex number that is$z = x + iy$.
$
   \Rightarrow 6 - 8i = x + iy \\
   \Rightarrow x = 6 \\
   \Rightarrow y = - 8 \\
 $
Step 2: Now, we have to find the value of $r$ with the help of the formula (A) which is mentioned in the solution hint.
$
   \Rightarrow r = \sqrt {{{\left( 6 \right)}^2} + {{\left( { - 8} \right)}^2}} \\
   \Rightarrow r = \sqrt {36 + 64} \\
   \Rightarrow r = \sqrt {100} \\
   \Rightarrow r = 10 \\
 $
Step 3: Now we have to find the value of $\theta $ with the help of the formula (B) which is mentioned in the solution hint.
$ \Rightarrow \theta = {\tan ^{ - 1}}\left( {\dfrac{{ - 8}}{6}} \right)$
Now, $6 - 8i$is in the 4th quadrant so we must ensure that $\theta $ is in the 4th quadrant.
$ \Rightarrow \theta = - {\tan ^{ - 1}}\left( {1.332} \right)$
Now, we know that $\tan \left( {0.927} \right) = 1.332$
$
   \Rightarrow \theta = - {\tan ^{ - 1}}\left[ {\left( {\tan 0.927} \right)} \right] \\
   \Rightarrow \theta = - 0.927 \\
 $
Step 3: Now, we have to find the value of $\cos \theta $and $\sin \theta $ as mentioned below.
$
   \Rightarrow \cos \theta = \cos \left( { - 0.927} \right) \\
   \Rightarrow \cos \theta = 0.6 \\
 $
And,
$
   \Rightarrow \sin \theta = \sin \left( { - 0.927} \right) \\
   \Rightarrow \sin \theta = - 0.8 \\
 $
Step 4: Now, the polar form of the given complex number $6 - 8i$ is in the form of $z = r(\cos \theta + i\sin \theta )$ that is $z = 10(0.6 - i0.8)$

Final solution: Hence, the polar form of the given complex number $6 - 8i$ is $10(0.6 - i0.8)$

Note:
It is necessary to understand about the complex number and the formulas which are mentioned in the solution hint.
It is necessary to check that the given complex number $6 - 8i$ lies in which quadrant.