e i theta and roots example
DESCRIPTION
theta roots exampleTRANSCRIPT
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Question (i) Show that, for any complex number ie ,
i
ie 1 1 i cot
2 2e 1
=
.
(ii) Write down the roots of the equation 5 1 0z = , leaving your answers in the form ie where 0r > and pi pi < .
(iii) Hence, solve ( )55 1w w= , leaving your answers in the form ix y+ where ,x y .
Solution
(i) i
i i 2
i i i2
e e e
e 1 e 1e
=
[ Multiply top and bottom by i2e
, since we want to obtain
an expression involving 2
. This is a good technique to use. ]
i2
i i2 2
e
e e
=
cos i sin2 2
2isin2
+
= [ It is useful to remember that i i2 2e e 2i sin
2
= and
that i i2 2e e 2cos
2
+ = . ]
cot1 i2 12 i i
= +
[ Simplify, especially the first term. ]
1 1 i cot2 2
=
(shown)
(ii) 2 i5 51 e
k
z zpi
= = , 0, 1, 2k = [ Complex roots occur in conjugate pairs. ]
(iii) ( )5
55 1 11
ww w
w
= =
[ Write it in a form like that of (ii). ]
1w [ Exclude any solution that will not satisfy the equation. ]
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Using (ii), 2 i
5e , 1, 21
kw k
w
pi
= =
[ 0k = gives 1w = , so it is excluded. ]
2 2i i5 5e e
k k
w w
pi pi
= [ We want to make w the subject on the LHS. ]
2 2i i5 5e 1 ek k
w
pi pi =
[ Rearrange and factorise the LHS. ]
2 i5
2 i5
e
e 1
k
kw
pi
pi=
[ Observe that the form looks like that of (i). ]
Using (i), 1 1 i cot , 1, 22 5
kw kpi = =
1 10.688i, 0.162i2 2
w = [ Express the imaginary parts to 3 significant figures. ]