Question

Find the cube root of $27(\cos {30}^o+i\sin {30}^o)$ that, when represented graphically, lies in the second quadrant.

• A. $3(\cos {10}^o+i\sin {10}^o)$
• B. $3(\cos {170}^o+i\sin {170}^o)$
• C. $3(\cos {100}^o+i\sin {100}^o)$
• D. $3(\cos {130}^o+i\sin {130}^o)$
• E. $3(\cos {150}^o+i\sin {150}^o)$

Answer 1

Answer: (D)

$3(\cos {130}^o+i\sin {130}^o)$

Given, $27(\cos 30+i\sin 30)=27e^{i(30+360n)}$, where $n$ is positive integer
The cube root of given one is equal to $3\left(e^{i(10+120n)}\right)$
Since they have given that the cube root lies in $2^{nd}$ quadrant, we get $n=1$
Which implies $3e^{i130}=3(\cos 130+i\sin 130)$