Complex number problems* (1 Viewer)

[norelle]

1) Given that arg(z-1) = 2pi/3 and arg(z+1) = pi/6, express z in mod-arg form.

2) Given that w = (z+2)/z and z moves along the unit circle. Find and describe the locus of w.


3) It is given that z1 and z2 are two complex numbers representing points A and B with
|z1|=|z2|=1, arg(z2)=2arg(z1) with 0<arg(z1)<pi/2

i) Show that arg(z2+1) = arg(z1)
ii) Show that |z2+1| = 2cos(argz1)

Please help~~~
THANKSSS

 

[Aerath]

These probably aren't the quickest ways of doing them, and I'm not sure they're right, but here you go:

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And I'm fairly stoned atm, so I can't do the very last question.

 

[Uncle]

oh shit as if complex numbers didnt annoy me enough in 1st year i have to make phasor diagrams.

 

[azureus88]

last question:

let A be arg(z1)

Using sine rule:

|z2 + 1|/sin(180-2A) = |z2|/sinA

|z2 + 1| = sin2A/sinA since |z2| = 1 and sin(180-2A)=sin2A

=2sinAcosA/sinaA
=2cosA
=2cos(arg(z1))

 

[norelle]

you guys are great`
thanks heapss

 

[gurmies]

An alternative solution to question 1:

Also, for question 3 part 1, you can use angle at the centre is twice angle at the circumference to show that alpha = beta.

 

[Aerath]

An alternative solution to question 1:

Argh - my eyes.

 

[GUSSSSSSSSSSSSS]

well if gurmies wants to post an alternate to q1 then so will i xD

the old sub in z = x + iy

arg(x - 1 + iy) = 2pi/3 .......... & arg(x + 1 + iy) = pi/6

since the argument is the tan inverse of the imaginary part over the real part these can be re written as:

y/(x - 1) = -sqrt3 ....................eqn1

y/(x + 1) = 1/sqrt3 .......................eqn2

eqn1/eqn2:

(x + 1)/(x - 1) = -3
x + 1 = -3x + 3
4x = 2
x = 1/2

sub into eqn 1

y/(-1/2) = -sqrt3
-2y = -sqrt3
y = (sqrt3)/2

and so complex number is
z = 1/2 + i(sqrt3)/2

and then put in mod/arg form xD

 

[gurmies]

Argh - my eyes.

Haha, pardon me - they recommended this size for it

 

[Puttah]

For Q2 I tried to solve it this way:

let
and

Then

After simplifying,

Equating real and imaginary parts: and

Can I go any further with this or was my entire setup just retarted?