Exercise 13.4
QUESTION 1
Find the modulus and argument of the following complex number and hence express each of them in the polar form :
(i)
Sol :
Let
Since point lies in the first quadrant , the argument of z is given by
Polar form
(ii)
Sol :
Let
Since point lies in the quadrant , the argument of z is given by
Polar form
(iii)
Sol :
Let
Since point lies in the quadrant , the argument of z is given by
Polar form
(iv)
Sol :
Rationalizing the denominator :
= 1
Since point lies on the negative direction of the imaginary axis , the argument of z is given by
Polar form
(v)
Sol :
Rationalizing the denominator
Let
= 1
Since point lies in the fourth quadrant , the argument is given by
Polar form
(vi)
Sol :
Rationalizing the denominator :
Let
= 1
Since point lies in the second quadrant , the argument is given by
Polar form
(vii)
Sol :
Let
Since point lies in the first quadrant , the argument is given by
Polar form
(viii)
Sol :
Rationalizing the denominator:
= 8
Let
Since the point lies in the third quadrant , the argument is given by
Polar form
QUESTION 2
Write in polar form
Sol :
*** QuickLaTeX cannot compile formula: \left[ \begin{array}{ll}{\because} & {i^{4}} & {=1}\end{array}\right] *** Error message: Extra alignment tab has been changed to \cr. leading text: ...ft[ \begin{array}{ll}{\because} & {i^{4}} &
Let
Then
Let be the argument of z and
be the acute angle given by
Then ,
Clearly, z lies in fourth quadrant. So,
The polar form of z is
Thus , the polar form of is
QUESTION 3
Express the following complex in the form
(i)
Sol :
Let
(ii)
Sol :
(iii)
Sol :
(iv)
Sol :