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Question: Find the modulus and amplitude of \(-\sqrt{3}-i\) \(\begin{aligned} & a)|z|=2;amp(z)=-\dfrac{...

Find the modulus and amplitude of −3−i-\sqrt{3}-i
a)∣z∣=2;amp(z)=−5π6 b)∣z∣=4;amp(z)=5π6 c)∣z∣=4;amp(z)=−π6 d) None of these \begin{aligned} & a)|z|=2;amp(z)=-\dfrac{5\pi }{6} \\\ & b)|z|=4;amp(z)=\dfrac{5\pi }{6} \\\ & c)|z|=4;amp(z)=-\dfrac{\pi }{6} \\\ & \text{d) None of these} \\\ \end{aligned}

Explanation

Solution

For a complex function of the form a + ib, we have the modulus of the complex number is a2+b2\sqrt{{{a}^{2}}+{{b}^{2}}} . Now to find the amplitude first we will find the value of tan⁡−1(ba){{\tan }^{-1}}\left( \dfrac{b}{a} \right) .

Complete step by step answer:
Now let us consider the given complex number.
The given complex number is −3−i-\sqrt{3}-i
Now we can also write this complex number as (−3)+(−1)i\left( -\sqrt{3} \right)+\left( -1 \right)i
Now comparing this equation with the general form of complex numbers which is a + bi, we get
a = −3-\sqrt{3} and b = −1-1 .
Let us first calculate the modulus of the complex number.
We know that modulus of the complex number of the form a + ib is a2+b2\sqrt{{{a}^{2}}+{{b}^{2}}}.
Hence we get the modulus of given complex number is (−3)2+(−1)2\sqrt{{{\left( -\sqrt{3} \right)}^{2}}+{{\left( -1 \right)}^{2}}}
(−3)2+(−1)2=3+1=4=2\sqrt{{{\left( -\sqrt{3} \right)}^{2}}+{{\left( -1 \right)}^{2}}}=\sqrt{3+1}=\sqrt{4}=2
Hence the modulus of a given complex number is 2.
Now let us find the argument of the given complex number.
To find the argument of any complex number of the form a + ib we first need to find tan⁡−1(ba){{\tan }^{-1}}\left( \dfrac{b}{a} \right)
Hence we get tan⁡−1(−1−3)=π6{{\tan }^{-1}}\left( \dfrac{-1}{-\sqrt{3}} \right)=\dfrac{\pi }{6} .
Now since we have 0<π6<π20<\dfrac{\pi }{6}<\dfrac{\pi }{2} .
Now since the point is in third quadrant the angle will be given by
π6−π=π−6π6=−5π6\dfrac{\pi }{6}-\pi =\dfrac{\pi -6\pi }{6}=-\dfrac{5\pi }{6} .
Hence the modulus of complex numbers is 2 and the amplitude is −5π6-\dfrac{5\pi }{6} .

So, the correct answer is “Option a”.

Note: Now note that the amplitude is not always equal to tan⁡−1(ba){{\tan }^{-1}}\left( \dfrac{b}{a} \right) . It depends on which quadrant, the complex number lies. For a complex number in the first quadrant the amplitude is tan⁡−1(ba){{\tan }^{-1}}\left( \dfrac{b}{a} \right) . If the complex number lies in the second quadrant then the amplitude is given by π−tan⁡−1(ba)\pi -{{\tan }^{-1}}\left( \dfrac{b}{a} \right) . Now is the point lies in third quadrant then the argument is tan⁡−1(ba)−π{{\tan }^{-1}}\left( \dfrac{b}{a} \right)-\pi and for fourth quadrant the amplitude is −tan⁡−1(ba)-{{\tan }^{-1}}\left( \dfrac{b}{a} \right) .