Mastering Mathematics Smartly www.freewebs.com/weews

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. ]

Mastering Mathematics Smartly www.freewebs.com/weews

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. ]