Question

How can you convert \[-729\] into polar form?

Answer 1

Hint: you should know about the complex number before solving this question. To convert a given complex number into its polar form, first, we will obtain the value by using a suitable formula and after this, we will find the angle using the suitable formula. Also, a complex number has both real and imaginary parts.

Complete step-by-step solution:
A number that can be expressed in the form of \[a+ib\] is known as a complex number. Where a and b are known as the real numbers and i is the symbol that is used to represent imaginary units.
The value of i is \[\sqrt{-1}\] and the square of i is \[-1\] and i is known as ‘iota’
Another way to represent a complex number is its polar form
The polar form of a complex number \[a+ib\] can be given as-
\[z=r(\cos \theta +i\sin \theta )\]……..(1)
Value of r can be found out by the given method –
If \[z=a+ib\]
Then, \[r=\sqrt{{{a}^{2}}+{{b}^{2}}}\]
Value of \[\theta \] can be obtained by the given method-
\[\begin{align}
  & a=r\cos \theta \\
 & b=r\sin \theta \\
\end{align}\]
If the value of a is greater than zero (\[a>0\]) then the value of \[\theta \] will be obtained with the below-given formula.
\[\theta ={{\tan }^{-1}}\left( \dfrac{b}{a} \right)\]
And if the value of a is less than zero \[(a<0)\] then the value of \[\theta \] will be obtained with the below-given formula.
\[\theta ={{\tan }^{-1}}\left( \dfrac{b}{a} \right)+{{180}^{{}^\circ }}\]
Now in the above question, it is asked to convert \[-729\] into its polar form
So \[-729\] can be written as
\[z=-729+0i\]
Where,
\[\begin{align}
  & a=-729 \\
 & b=0 \\
\end{align}\]
Now we have to represent the complex number \[z=-729+0i\] into polar form. So for that first, we will obtain the value of r which will be obtained as follows
\[r=\sqrt{{{(-729)}^{2}}+{{0}^{2}}}\]
Value of r will come out to be
\[r=729\]
In this question, the value of a is less than zero, so the value of \[\theta \] can be given by
\[\theta ={{\tan }^{-1}}\left( \dfrac{0}{-729} \right)+{{180}^{{}^\circ }}\]
\[\Rightarrow \theta ={{\tan }^{-1}}(0)+\pi \]
As the value of \[{{\tan }^{-1}}(0)=0\], therefore
value of \[\theta =\pi \]
Now putting the values of r and \[\theta \] in eq(1), we obtain the following results
\[z=729(\cos \pi +i\sin \pi )\]
So this is the required answer to the above question

Note: It should be noted that the sum of any two conjugate complex numbers is always a real number and also the product of any two conjugate complex numbers is also a real number. If both the product as well as the sum of two complex numbers are real then the complex number is the conjugate of each other.