Compute the modulus and argument of each complex number?
a.
\(\sqrt{0+25} =5\)
b.
\(\sqrt{ cos^{2} 5pi/7 +sin^{2} 5pi/7 } = \sqrt{1} =1\)
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Answer and Explanation:
To find : Compute the modulus and argument of each complex number?
Solution :
a) \(-5i\)
When the complex number is in the form \(z=a+ib\)
Then the modulus of z is \(|z|=\sqrt{a^2+b^2}\) and argument is \(\theta=\tan^{-1}(\frac{b}{a})\)
On comparing with given complex number, a=0 and b=-5
Substitute in formulas,
The modulus of z is \(|z|=\sqrt{0^2+(-5)^2}=\sqrt{25}=5\)
The argument of z is \(\theta=\tan^{-1}(\frac{-5}{0})=\tan^{-1}(\tan(90))=90^\circ\)
b) \(\sqrt{10}(\cos(\frac{5\pi}{7})+i\sin(\frac{5\pi}{7}))\)
When the complex number is in the polar form \(z=r(\cos\theta+i\sin\theta)\)
Then the modulus of z is r and argument is \(\theta\).
On comparing with given complex number,
The modulus of z is \(r=\sqrt{10}\)
The argument of z is \(\theta=\frac{5\pi}{7}\)