# Thread: Express the eq in a+ib form

1. ## Express the eq in a+ib form

Express $z = \frac{(cos \theta + i sin \theta)^3(sin \theta + i cos \theta)}{(cos2 \theta - i sin2 \theta)}$ in the form $a+ib$

Solution :

$z = \frac{(cos \theta + i sin \theta)^3(sin \theta + i cos \theta)}{(cos2 \theta - i sin2 \theta)}$

$= \frac{(cos \theta + i sin \theta)^3(sin \theta + i cos \theta)}{(cos \theta + i sin \theta)^{-2}}$

$= (cos \theta + i sin \theta)^5(sin \theta + i cos \theta)$

Stuck !!!!!

2. Originally Posted by TomJerry
Express $z = \frac{(cos \theta + i sin \theta)^3(sin \theta + i cos \theta)}{(cos2 \theta - i sin2 \theta)}$ in the form $a+ib$

Solution :

$z = \frac{(cos \theta + i sin \theta)^3(sin \theta + i cos \theta)}{(cos2 \theta - i sin2 \theta)}$

$= \frac{(cos \theta + i sin \theta)^3(sin \theta + i cos \theta)}{(cos \theta + i sin \theta)^{-2}}$

$= (cos \theta + i sin \theta)^5(sin \theta + i cos \theta)$

Stuck !!!!!
Assuming your last line is correct (and I haven't bothered to check) then you can use deMoivre's Theorem to write $( \cos \theta + i \sin \theta )^5 = \cos (5 \theta) + i \sin (5 \theta)$. Now expand $(\cos (5 \theta) + i \sin (5 \theta)) (\cos ( \theta) + i \sin (\theta))$ and group the real and imaginary parts.

3. Originally Posted by mr fantastic
Assuming your last line is correct (and I haven't bothered to check) then you can use deMoivre's Theorem to write $( \cos \theta + i \sin \theta )^5 = \cos (5 \theta) + i \sin (5 \theta)$. Now expand $(\cos (5 \theta) + i \sin (5 \theta)) (\cos ( \theta) + i \sin (\theta))$ and group the real and imaginary parts.
U are missing the question if u look closely the last line is

$(cos 5\theta + i sin 5\theta) (sin \theta + i cos \theta)$

and not as u have mentioned it

$(cos 5 \theta + i sin 5\theta) (cos \theta + i sin \theta)$

4. Originally Posted by TomJerry
U are missing the question if u look closely the last line is

$(cos 5\theta + i sin 5\theta) (sin \theta + i cos \theta)$

and not as u have mentioned it

$(cos 5 \theta + i sin 5\theta) (cos \theta + i sin \theta)$
You should realise that my small error has absolutely no affect on the advice I have given you.

5. sin (theta)=cos((pi)/2-(theta))
cos(theta)=sin((pi)/2-(theta))

6. Originally Posted by mr fantastic
You should realise that my small error has absolutely no affect on the advice I have given you.
how will u group the real parts when the one is a cos and the other is a sin ..........

7. Originally Posted by TomJerry
how will u group the real parts when the one is a cos and the other is a sin ..........
Can you not expand $(\cos (5\theta) + i \sin (5\theta)) (\sin \theta + i \cos \theta)$ and then group together the real and imaginary bits??

8. Originally Posted by mr fantastic
Can you not expand $(\cos (5\theta) + i \sin (5\theta)) (\sin \theta + i \cos \theta)$ and then group together the real and imaginary bits??
could u please show me a example or even a link would be more help.......i just need to know how to carry out this .......different signs

9. Originally Posted by TomJerry
could u please show me a example or even a link would be more help.......i just need to know how to carry out this .......different signs
$(\cos{5\theta} + i\sin{5\theta})(\sin{\theta} + i\cos{\theta}) = \cos{5\theta}\sin{\theta} + i\cos{5\theta}\cos{\theta} + i\sin{5\theta}\sin{\theta} + i^2\sin{5\theta}\cos{\theta}$

$= \cos{5\theta}\sin{\theta} - \sin{5\theta}\sin{\theta} + i(\cos{5\theta}\cos{\theta} + \sin{5\theta}\sin{\theta})$.

What do you think $a$ and $b$ are?

10. Here are some short cuts.
$\frac{{\left[ {\cos (\varphi ) + i\sin (\varphi )} \right]^3 \left( {\sin (\varphi ) + i\cos (\varphi )} \right)}}{{\cos (2\varphi ) - i\sin (2\varphi )}}$ $= \left[ {\cos (3\varphi ) + i\sin (3\varphi )} \right]\left( {\sin (\varphi ) + i\cos (\varphi )} \right)\left( {\cos (2\varphi ) + i\sin (2\varphi )} \right)$
$\left( {\sin (\varphi ) + i\cos (\varphi )} \right) = \left( {\cos (\varphi ) - i\sin (\varphi )} \right)i = \left( {\cos ( - \varphi ) + i\sin ( - \varphi )} \right)i$

11. thanks