14. Suppose that the complex number z has modulus one, and that 0 < Arg z < . Prove that 2 Arg(z + 1) = Arg z.

Algebra & Trigonometry with Analytic Geometry
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Chapter3: Functions And Graphs
Section3.3: Lines
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14. Suppose that the complex number z has modulus one, and that 0 < Arg z < 1.
Prove that 2 Arg(z + 1) = Argz.
15. The vertices of the quadrilateral ABCD in the complex plane represent the complex
numbers 2₁, 22, 23 and 24 respectively.
(a) If 2₁ - 22 = 24 - 23, show that the quadrilateral ABCD is a parallelogram.
(b) If 2₁ - 22 =24 — 23 and 2₁ — 23 = i(za - 22), show that ABCD is a square.
16. A triangle has vertices at points in the Argand diagram which
represent the complex numbers z₁, 22 and 23.
Z2₂ - 21
23 - 21
If
= cos+ i sin, show that the triangle is equilateral.
17. In the diagram on the right, the points P₁, P₂ and P3 represent
Z2 Z3
the complex numbers z₁, 22 and 23 respectively. If
show
21
that OP₂ bisects LP₁OP3.
=
22
"
•P3
•P₂
•P₁
Transcribed Image Text:14. Suppose that the complex number z has modulus one, and that 0 < Arg z < 1. Prove that 2 Arg(z + 1) = Argz. 15. The vertices of the quadrilateral ABCD in the complex plane represent the complex numbers 2₁, 22, 23 and 24 respectively. (a) If 2₁ - 22 = 24 - 23, show that the quadrilateral ABCD is a parallelogram. (b) If 2₁ - 22 =24 — 23 and 2₁ — 23 = i(za - 22), show that ABCD is a square. 16. A triangle has vertices at points in the Argand diagram which represent the complex numbers z₁, 22 and 23. Z2₂ - 21 23 - 21 If = cos+ i sin, show that the triangle is equilateral. 17. In the diagram on the right, the points P₁, P₂ and P3 represent Z2 Z3 the complex numbers z₁, 22 and 23 respectively. If show 21 that OP₂ bisects LP₁OP3. = 22 " •P3 •P₂ •P₁
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