Polar form of a complex number solutions

 
Polar form of a complex number. If z= x + iy is a complex number. Then z = r ( cosθ + sinθ ) is called polar form or trigonometric form of a complex number.
 
By comparing real and imaginary parts of a complex number,  we get
x = r cosθ           (1)
y = r sinθ           (2)
 
By squaring above equations and adding we get
r^2 = x^2 +y^2
 
 
 
EXAMPLE
 
 
 
Express -\sqrt{3} -i in polar form.
 
 
Solution
 
With x =-\sqrt{3} and y = −1 we obtain
 
r = |z| =\sqrt{(\sqrt{-3})^2+ (-1)^2}= 2
Now
 
\frac{y}{x} = \frac{-1}{-\sqrt{3}}=\frac{1}{\sqrt{3}}\
and
\theta=tan^{-1}(\frac{y}{x})=\frac{1}{\sqrt{3}}= \frac{\pi}{6}\
 
 which is an angle whose terminal side is in the first quadrant.But
 
since the point (-\sqrt{3}, -1) lies in the third quadrant, we take the solution
tan \theta =\frac{-1}{\sqrt{-3}}=\frac{1}{\sqrt{3}}
So,
\theta = arg(z) = \frac{\pi}{6}+ \pi = \frac{7\pi}{6}
Therefore, polar form of the number is
z = 2(cos\frac{7\pi}{6} + isin \frac{7\pi}{6})
 
Solved Questions of polar form
CHALLENGE
 
write the given complex number in polar form first using an argument θ = Arg(z)
1.       2
2.    −10
3.     −3i
4.       6i
5.       1 + i
6.       5 − 5i
8.    -2 - 2\sqrt{3i}
9.    \frac{3}{-1 + i}
10.  \frac{12}{\sqrt{3} + i}
 
Note: If you failed to find any question solution then comments us. We’ll provide you their solutions.
 
 

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