Question

The principal argument of $z=-3+3i$ is:
(a) $\dfrac{\pi }{4}$
(b) $-\dfrac{\pi }{4}$
(c) $\dfrac{3\pi }{4}$
(d) $-\dfrac{3\pi }{4}$

Answer 1

Hint: To find the principal argument of the given complex number, we have to first of all find the argument of the given complex number. Argument of a complex number is the inverse of tan of the ratio of the imaginary number to the real number. And the principal argument is the angle which lies between 0 to $2\pi $.

Complete step by step solution:
The complex number given in the above problem is as follows:
$z=-3+3i$
The real part of the above complex number is -3 and the imaginary part of the above complex number is 3. Now, we are going to divide the imaginary part to the real part and we get,
$\dfrac{3}{-3}$
In the above expression, 3 will be cancelled out from the numerator and the denominator and we get,
$-1$
Now, the above value is equal to $\tan \theta $ so equating the above value to $\tan \theta $ we get,
$\tan \theta =-1$
The $\theta $ in the above equation is the argument of the complex number and as argument should be principal argument so $\theta $ must lie from $-\dfrac{\pi }{2}to\dfrac{\pi }{2}$ . Now, we know that $\theta $ at which tan is -1 is $-\dfrac{\pi }{4}$ but our answer should lie from $-\dfrac{\pi }{2}to\dfrac{\pi }{2}$ so we have to add $\pi $ to $-\dfrac{\pi }{4}$ and we get,
$\pi -\dfrac{\pi }{4}$
Taking 4 as L.C.M in the above expression we get,
$\dfrac{4\pi -\pi }{4}=\dfrac{3\pi }{4}$
From the above, we got the principal argument of the complex number $z=-3+3i$ as $\dfrac{3\pi }{4}$.

So, the correct answer is “Option (c)”.

Note: The mistake that could be possible in the above problem is that you might forget to convert the argument into principal argument and interestingly you will get that angle in the option also which is the option (b) and you will happily tick this incorrect option so make sure you won’t make this mistake.