Question 1: Find the modulus and argument of the following complex numbers and hence express each one of them in polar form:
Answer:
Since the point lies in the first quadrant, the
is given by
Since the point lies in the first quadrant, the
is given by
Since the point lies in the fourth quadrant, the
is given by
Since the point lies on the negative direction of the imaginary axis, the
is given by
Since the point lies in the fourth quadrant, the
is given by
Since the point lies in the second quadrant, the
is given by
Since the point lies in the first quadrant, the
is given by
Since the point lies in the third quadrant, the
is given by
Question 2: Write in polar form
Answer:
Since the point lies in the fourth quadrant, the
is given by
Question 3: Express the following complex numbers in the form :
Answer:
is a periodic function with period
.
Hence we take
We can see that ,
lies in first quadrant,
.
Thus in the polar form is given by
,
We can see that ,
lies in fourth quadrant,
.
Thus in the polar form is given by
is a periodic function with period
.
Hence we take
We can see that ,
lies in fourth quadrant, therefore
.
Thus in the polar form is given by
,
We can see that ,
lies in third quadrant, therefore
.
Thus in the polar form is given by
Both Sine and Cosine functions are periodic function with period
Hence let us take
In this case
lies in the first quadrant, therefore
Hence the is
In this case
lies in the fourth quadrant, therefore
Hence the is
In this case
lies in the first quadrant, therefore
Hence the is
lies in the fourth quadrant, therefore
Hence the is
Question 4: If are two complex numbers such that
, then show that
Answer:
by
by
;
is a complex number
Question 5: If are two pairs of conjugate complex numbers, prove that
Answer:
are two pairs of conjugate complex numbers.
… … … … … i)
… … … … … ii)
. Hence proved.
Question 6: Express in polar form.
Answer: