suppose z1,z2,z3 are the vertices of an equilateral triangle inscribing the circle mod z=1 If z1=1+root3i,find z2 and z3 - Maths - Permutations and Combinations

Question

suppose z1,z2,z3 are the vertices of an equilateral triangle
inscribing the circle mod z=1 If z1=1+root3i,find z2 and z3

MD. Akram · Accepted Answer

Dear Student! We assume that your equation of circle is Mod(z) =2 Beacuse mod z =1 represent a circle with center (0,0) and radious 1 And the distance between (0,0) and z1 become 2 which is grater then 1 Ans: $| z | = 2$ represents a circle with centre (0, 0) and radius = 2 z1, z2 and z3 are the vertices of an equilateral triangle inscribed in the given circle Given, $z_{1} = 1 + \sqrt{3} i$

$z_{1} = 1 + \sqrt{3} i = 2 (\frac{1}{2} + \frac{\sqrt{3}}{2} i) = 2 (\cos \frac{π}{3} + i \sin \frac{π}{3}) = 2 e^{\frac{i π}{3}}$ $z_{2} = z_{1} e^{i \frac{2 π}{3}} = 2 e^{i \frac{π}{3}} \cdot e^{i \frac{2 π}{3}} = 2 e^{π i} = 2 (\cos π + i \sin π) = 2 (- 1 + i \times 0) = - 2$ $z_{3} = z_{1} e^{i \frac{4 π}{3}} = 2 e^{i \frac{π}{3}} \cdot e^{i \frac{4 π}{3}} = 2 e^{\frac{5 π}{3} i} = 2 [\cos (\frac{5 π}{3}) + i \sin (\frac{5 π}{3})] = 2 [\cos (2 π - \frac{π}{3}) + i \sin (2 π - \frac{π}{3})] = 2 (\cos \frac{π}{3} - i \sin \frac{π}{3}) = 2 (\frac{1}{2} - i \frac{\sqrt{3}}{2}) = 1 - \sqrt{3} i$ Thus, $z_{2} = - 2 a n d z_{3} = 1 - \sqrt{3} i$ Cheers!

suppose z1,z2,z3 are the vertices of an equilateral triangle inscribing the circle mod z=1. If z1=1+root3i,find z2 and z3