Question

A root of ${x^5} - 32 = 0\;$ lies in quadrant II. Write this root in polar form.
$2\left( {\cos {{120}^0} + i\sin {{120}^0}} \right)$
A. $2\left( {\cos {{144}^0} + i\sin {{144}^0}} \right)$
B. $2\left( {\cos {{150}^0} + i\sin {{150}^0}} \right)$
C. $4\left( {\cos {{144}^0} + i\sin {{144}^0}} \right)$
D. $2\left( {\cos {{72}^0} + i\sin {{72}^0}} \right)$

Answer 1

To express the root of the given equation in polar form, first of all solve the equation for $x$ and then express it in Euler’s form. And after expressing the solution in Euler’s form, find the angle for the solution such that the angle will satisfy the criteria given in the question and then just convert the solution into polar form.
Euler form to polar form is converted as
$A{e^{i\theta }} = A\left( {\cos \theta + i\sin \theta } \right)$

Complete step by step solution:
In order to express the root of the given equation ${x^5} - 32 = 0\;$ which is lying on the second quadrant, we will first find and convert the root of the given equation in Euler form, as follows
We can write the given equation as

$$\Rightarrow {x^5} - 32 = 0\; \\\ \Rightarrow {x^5} = 32 \\\$$

Using law of indices for multiplication, we can write $32 = {2^5}$
$\Rightarrow {x^5} = {2^5}$
Now, from Euler’s formula we know that, $A{e^{i\theta }} = A\left( {\cos \theta + i\sin \theta } \right)$, using this simplifying the above equation further, we will get
$\Rightarrow {x^5} = {2^5}{e^{i2\pi n}},\,n \in I$
Taking fifth root, both sides, we will get
$\Rightarrow x = 2{e^{\dfrac{{i2\pi n}}{5}}},\,n \in I$
Now, as given in the question, the root is lying in the second quadrant, so we have to find the value of $n$ such that we will get an angle in the interval $\left[ {\dfrac{\pi }{2},\,\pi } \right]$
On putting $n = 2,\;\theta = \dfrac{{4\pi }}{5}$
$\Rightarrow x = 2{e^{i\dfrac{{4\pi }}{5}}}$
Converting the angle in degrees and the root in polar form,
$\Rightarrow x = 2\left( {\cos {{144}^0} + i\sin {{144}^0}} \right)$

Hence, the option (B) is the correct answer.

Note: When converting from Euler’s into polar form, take care of the sign of the angle (if negative exists) and then convert the equation accordingly. Also write your answer in the units as given in options before selecting your answer, because it happens that one has solved the question correctly but selected the wrong option.