Question

How can you convert $-729$ into polar form?

Answer 1

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-

& a=r\cos \theta \\\ & b=r\sin \theta \\\ \end{aligned}$$ If the value of a is greater than zero ($$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{aligned} & a=-729 \\\ & b=0 \\\ \end{aligned}$$ Now we have to represent the complex number $$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.