complex number question (1 Viewer)

[poptarts12345]

how do i solve this :

Suppose that the complex number z has modulus one, and that 0 < Arg z < π/2 . Prove that 2 Arg(z + 1) = Arg z.

 

[Drongoski]

Geometrically, draw a unit circle, pick any point on the circle in quadrant-1, to represent z. The complex number 1 is on the circle where it intersects the x-axis. If you draw the parallelogram with adjacent sides representing z and 1 you have the diagonal thru the origin as the complex number z+1. This diagonal bisect the angle of the //gram at the origin (this is arg(z)) since it is a rhombus. Therefore this diagonal bisects said angle. But arg(z+1) is half of this angle. That is it is half of arg(z). Hence arg(z) = 2 x arg(z+1)

 

[erucibon]

Plot out the info on a unit circle and use a circle geo property

 

[ultra908]

r we expected to know circle geo properties for questions like this

Edit: no circle geo required for this question. But are we expected to know certain circle geo theorems to do complex questions? ik some schools do circle geo in yr10 but mine didnt. If we do need to know them, which ones?

 

[erucibon]

r we expected to know circle geo properties for questions like this
Edit: no circle geo required for this question. But are we expected to know certain circle geo theorems to do complex questions? ik some schools do circle geo in yr10 but mine didnt. If we do need to know them, which ones?

It is supposed to be assumed knowledge, but it is just the one where the angle at the circumference is half the angle at the centre. That one is the main one you need for some locus questions in 4u

 

[ultra908]

It is supposed to be assumed knowledge, but it is just the one where the angle at the circumference is half the angle at the centre. That one is the main one you need for some locus questions in 4u

Which circle geo theorems r assumed knowledge?

 

[mani88]

how do i solve this :

Suppose that the complex number z has modulus one, and that 0 < Arg z < π/2 . Prove that 2 Arg(z + 1) = Arg z.

You can also do it algebraically by considering

 

[erucibon]

Which circle geo theorems r assumed knowledge?

Technically all of them that other people learnt in year 10, but the one i mentioned is the only one I've used in yr 12 (if i can remember corectly

 

[ultra908]

Technically all of them that other people learnt in year 10, but the one i mentioned is the only one I've used in yr 12 (if i can remember corectly

lmao do u have a list of those theorems or somewhere i can find them? ty

 

[erucibon]

here is a list, but now that i actually look at them you honestly only need to know the circum-centre angle one and that tangents are perpendicular to centre

 

[ultra908]

tysm

 

[Drongoski]

Hardly any circle geometry involved.

 

[beetree1]

My teacher told me we arent allowed to use circle geometry in proving anymore since its not in syllabus? SOMEONE CONFIRM??

 

[idkkdi]

My teacher told me we arent allowed to use circle geometry in proving anymore since its not in syllabus? SOMEONE CONFIRM??

seems like bullsht that nesa and external markers would do. Don't they get paid per paper? If it's paid per paper, honestly, they wouldn't bother with things that look different.

 

[erucibon]

My teacher told me we arent allowed to use circle geometry in proving anymore since its not in syllabus? SOMEONE CONFIRM??

It is assumed knowledge and indirectly examinable