P4 Complex number Q

Find the modulus and the argument in radians in terms of pi of:
a) z1 = (1+i)(1-i).
Done: modulus = 1, arg = pi/2 radians
b) z2 = (root2)/(1-i)
Done: Modulus = 1, arg = pi/4 radians
Plot z1, z2 and z1 + z2 on a argand diagram: Done.
Here's the part where i'm stuck with: deduce tan(3pi/8) = 1 + root2

Any help is much appreciated.

Reply 1

Jonny W

Please look here: http://www.thestudentroom.co.uk/t63700.html.

Reply 2

Gaz031

Thanks for the link+solution.

Reply 3

Gaz031

I don't get:

2) Given that z = cos x + isin x show that:

2/(1 + z) = 1 - itan (x/2)

Could you elaborate for me? God i hope i haven't made the wrong choice choosing to do maths at uni.

Reply 4

Gaz031

I've got to:

[2sin^2 (.5theta) - isin(theta)] / [1-cos^2theta - isinthetacosteta]

Reply 5

Jonny W

2/(1 + z)
= 2 / (1 + cos x + i sin x)
= 2 (1 + cos x - i sin x) / [(1 + cos x)^2 + sin^2 x]
= 2 (1 + cos x - i sin x) / [1 + 2cos x + cos^2 x + sin^2 x]
= 2 (1 + cos x - i sin x) / [2 + 2cos x]
= 1 - i sin x / [1 + cos x]
= 1 - 2i sin(x/2) cos(x/2) / [2 cos^2(x/2)]
= 1 - i tan(x/2).

Reply 6

Gaz031

Thanks

2/(1 + z)
= 2 / (1 + cos x + i sin x)
= 2 (1 + cos x - i sin x) / [(1 + cos x)^2 + sin^2 x]
= 2 (1 + cos x - i sin x) / [1 + 2cos x + cos^2 x + sin^2 x]
= 2 (1 + cos x - i sin x) / [2 + 2cos x]
[Where did the cosx from the numerator go on the line below]
= 1 - i sin x / [1 + cos x]
= 1 - 2i sin(x/2) cos(x/2) / [2 cos^2(x/2)]
= 1 - i tan(x/2)

Reply 7

Fermat

2/(1 + z)
= 2 / (1 + cos x + i sin x)
= 2 (1 + cos x - i sin x) / [(1 + cos x)^2 + sin^2 x]
= 2 (1 + cos x - i sin x) / [1 + 2cos x + cos^2 x + sin^2 x]
= 2 (1 + cos x - i sin x) / [2 + 2cos x]
[Where did the cosx from the numerator go on the line below]
= 2(1+ cosx) / [2 + 2cos x] - 2i sinx / [2 + 2cos x]
= 1 - i sin x / [1 + cos x]
= 1 - 2i sin(x/2) cos(x/2) / [2 cos^2(x/2)]
= 1 - i tan(x/2)

Hope the extra line clears things up.

Reply 8

Jonny W

2 (1 + cos x - i sin x) / [2 + 2cos x]
= (1 + cos x - i sin x) / [1 + cos x]
= (1 + cos x) / [1 + cos x] - i sin x / [1 + cos x]
= 1 - i sin x / [1 + cos x]