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john1464 · Nov 4, 2015

This is question 5 in exercise 2.3 from the Cambridge

5. on an Argand diagram the points p and q represent the numbers z1 and z2 respectively. OPQ is an equilateral triangle. Show that z1^2+zx^2=z1z2

Can I use cosine rule to prove this or do I need to use the worked solutions method in which you let a=cispi/3 then at some point they write (1+a^2)=a which i dont understand if anyone could explain that step?

InteGrand · Nov 5, 2015

john1464 said:
This is question 5 in exercise 2.3 from the Cambridge

5. on an Argand diagram the points p and q represent the numbers z1 and z2 respectively. OPQ is an equilateral triangle. Show that z1^2+zx^2=z1z2

Can I use cosine rule to prove this or do I need to use the worked solutions method in which you let a=cispi/3 then at some point they write (1+a^2)=a which i dont understand if anyone could explain that step?

$Yes, you can use the cosine rule if you want. Also, for the solution's method, if $a=\mathrm{cis} \left( \frac{\pi}{3}\right)$, note that then $a^3 = \mathrm{cis} \left(\pi\right)= -1$ and $a\neq -1$, so $a^3 + 1 = 0 \Rightarrow (a+1)(a^2 - a + 1) = 0 \Rightarrow a^2 - a + 1 = 0$, as $a\neq -1$. Rearranging this last equation gives $a=1+a^2$.$

Need help with a question (2 Viewers)

john1464

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InteGrand

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