127.[HSC] (i) (ii) The complex numbers z = cos 0 + i sin 0 and w = cos α + isinα, -π<0≤π and -π<α ≤π, satisfy 1+z+w = 0. By considering the real and imaginary parts of 1+z+w, or otherwise, show that 1,z and w from the vertices of an equilateral triangle in the Argand diagram. Hence, or otherwise, show that if the three non-zero complex numbers 2i, Z₁ and Z₂ satisfy |2i| = |Z₁| = |Z₂ AND 2i+Z₁ + Z2₂ = 0 then they form the vertices of an equilateral triangle in the Argand diagram
127.[HSC] (i) (ii) The complex numbers z = cos 0 + i sin 0 and w = cos α + isinα, -π<0≤π and -π<α ≤π, satisfy 1+z+w = 0. By considering the real and imaginary parts of 1+z+w, or otherwise, show that 1,z and w from the vertices of an equilateral triangle in the Argand diagram. Hence, or otherwise, show that if the three non-zero complex numbers 2i, Z₁ and Z₂ satisfy |2i| = |Z₁| = |Z₂ AND 2i+Z₁ + Z2₂ = 0 then they form the vertices of an equilateral triangle in the Argand diagram
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![127.[HSC]
(i)
(ii)
The complex numbers z = cos 0 + i sin 0 and w = cos α + isinα, -π<0≤π and -π<α ≤π,
satisfy 1+z+w = 0. By considering the real and imaginary parts of 1+z+w, or otherwise, show that
1,z and w from the vertices of an equilateral triangle in the Argand diagram.
Hence, or otherwise, show that if the three non-zero complex numbers 2i, Z₁ and Z₂ satisfy
|2i| = |Z₁| = |Z₂ AND 2i+Z₁ + Z2₂ = 0
then they form the vertices of an equilateral triangle in the Argand diagram](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F41283726-d1bd-4b07-9b28-b2b15bfbf6be%2F8444e75f-e032-4586-9659-292f7d78dfe8%2Fx5q12li_processed.jpeg&w=3840&q=75)
Transcribed Image Text:127.[HSC]
(i)
(ii)
The complex numbers z = cos 0 + i sin 0 and w = cos α + isinα, -π<0≤π and -π<α ≤π,
satisfy 1+z+w = 0. By considering the real and imaginary parts of 1+z+w, or otherwise, show that
1,z and w from the vertices of an equilateral triangle in the Argand diagram.
Hence, or otherwise, show that if the three non-zero complex numbers 2i, Z₁ and Z₂ satisfy
|2i| = |Z₁| = |Z₂ AND 2i+Z₁ + Z2₂ = 0
then they form the vertices of an equilateral triangle in the Argand diagram
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