Q on argand diagram- confused about part iii. Why is it sqrt 2/2 x 2/sqrt 2?

[Sonal7]

 

[Sonal7]

The question says to use part II to work out the exact values of cos 7/12 pie and sin 7/12 pie.

 

[pka]

The question says to use part II to work out the exact values of cos 7/12 pie and sin 7/12 pie.

It is not pie but is pi.
\(\displaystyle \frac{{{z_1}}}{{{z_2}}} = \frac{{ - 1 + i}}{{\sqrt 3 + i}} = \frac{{1 - \sqrt 3 }}{4} + i\frac{{1 + \sqrt 3 }}{4}\)

\(\displaystyle \cos \left( {\frac{{7\pi }}{{12}}} \right) = \operatorname{Re} \left( {\frac{{{z_1}}}{{{z_2}}}} \right) = \frac{{1 - \sqrt 3 }}{4}~\&~\sin \left( {\frac{{7\pi }}{{12}}} \right) = \operatorname{Im} \left( {\frac{{{z_1}}}{{{z_2}}}} \right) = \frac{{1 + \sqrt 3 }}{4}\)

 

[Dr.Peterson]

View attachment 14487

If you are asking why they wrote [MATH]\frac{\sqrt{2}}{2}\times\frac{2}{\sqrt{2}}[/MATH] in the first step for part (iii), that is (I think) because they want to write the result of part (ii) in polar form, as a magnitude r times a complex number with magnitude 1. Since the magnitude of the number from (ii) is [MATH]\frac{\sqrt{2}}{2}[/MATH] (show that!), that is the r they need, and the number divided by [MATH]\frac{\sqrt{2}}{2}[/MATH] has magnitude 1. That is what they have on the last line you showed. What they did was to multiply the number by [MATH]r \times \frac{1}{r}[/MATH] in order not to change the number's value.

From there, they do what pka demonstrated, to finish the work.

 

[pka]

To Sonal7, this is one of the most useful identities one can know: \(\displaystyle \dfrac{1}{z}=\dfrac{\overline{z}}{|z|^2}\)
So that \(\displaystyle \dfrac{z_1}{z_2}=\dfrac{z_1\overline{z_2}}{|z_2|^2}\) can you use this?