Example 7
Find the continued product of all values of
Solution
1 +
= cos (2 p k + p
/ 3) + sin (2 p k + p
/ 3)
Note that r =
= 1 and q = tan^{1}(Ö
3 / 2) = 60^{0} or p/3^{c}
putting k = 0, 1, 2, 3, we get
Therefore the continued product is
\ z_{1} . z_{2} . z_{3}
. z_{4} = cos (10p) + i sin (10p)
= cos (0) + i sin (0) = 1 + i (0) = 1
Example 8
If l + 2i is one root of x^{4}  3x^{3} + 8x^{2}
 7x +5 = 0. Find the other roots.
Solution
Let a = = 1 + 2i then b
= 1 2i
We know that a quadratic equation in x, having roots a
and b, can be put in the form
x^{2}  ( a + b ) x + a . b = 0
\ x^{2}  [ (1 + 2i) + (1
 2i) ] x + (1 + 2i) (1  2i) = 0
\ x^{2}  2x + 5 = 0
Dividing by x^{2}  2x + 5 = 0 to x^{4}  3x^{3}
+ 8x^{2}  7x + 5 = 0 as
Since R = 0, x^{2}  x + 1 = 0 is the other factor of
the given expression.
Thus the other factor is x^{2}  x + 1 whose roots are
Now x =
Therefore x^{4}  3x^{3} + 8x^{2} 
7x + 5 = 0 has roots
1 + 2i, 1  2i,
Example 9
Find two values of
Solution
i = cos 90^{0} + i sin 90^{0}....[ ^{ }since
cos 90^{0} = 0 and sin 90^{0} = 1]
Example 10 Find the two values
of the sq.root of
Solution
1 + 2i Þ a = 1 > 0 and b =
2 > 0 i.e. (q is in 1st quad.)
\ r =
= 63^{0}26^{'}
Also, 2 + i Þ a = 2 > 0 and
b = 1 > 0 i.e. (q is in 1st quad.)
\ r = Ö4+1
= Ö5 and q
= tan^{1}(1/2) = 26^{0}34^{'}
\
= cos (63^{0}24^{'}  26^{0}36') + i sin
(63^{0}24'  26^{0}36' )
= cos (36^{0}52') + i
sin (36^{0}52')
Now
= cos 36^{0}52' + 2pk + i
sin 2pk + 36^{0}52'
....... by using z^{1/n} = r^{1/n}
When k = 0
\
= cos (18^{0}26') + i sin (18^{0}26') = 0.9487 +
0.3162i
When k = 1
= cos (198^{0}26') + i sin (198^{0}26')
=  0.9487  0.3162i
Thus required sq.roots are ± (0.9487 + 0.3162i)
Example 11 The centre of a regular hexagon is at the origin
and one vertex is given by (1+i) on the Argand's diagram. Find the
remaining vertices.
Solution
Since
represent Aº (1+ i) the first vertex
of the regular hexagon ABCDEF.
Now 1+ i Þ a=1 and b=1 (i.e. q
in 1st quad.). We have r=
=
units i.e. 
=
units and q = tan^{1}(1/1) =
p/4 = 45^{0}. \
makes 45^{0} angle with
and of length
units. Being regular hexagon ,
,
,
,
q make angles of p/3,
2p/3, p, 4p/3,
5p/3 with .
This shows that modulus of all these six vertices is
units and their respective amplitudes are, (p/4+
p/3), (p/4+2p/3),
(p/4+p), (p/4+5p/3)
for B, C, D, E and F.
\ Radius vectors representing the vertices
are
A,
B,
C,
D,
E,
and
F,
i.e. A is (1+i), B is (0.366 + 1.366i), C is (1.366 + 0.366i),
D is (1i), E is (0.366  1.366i) and F is (1.366  0.366i) .
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