4.02f Convert between forms: cartesian and modulus-argument

7

1. 1. Find the modulus and argument of \(z _ { 1 }\), where \(z _ { 1 } = 1 + \mathrm { i }\).
   2. Given that \(\left| z _ { 2 } \right| = 2\) and \(\arg \left( z _ { 2 } \right) = \frac { 1 } { 6 } \pi\), express \(z _ { 2 }\) in a + bi form, where \(a\) and \(b\) are exact real numbers.
2. Using these results, find the exact value of \(\sin \frac { 5 } { 12 } \pi\), giving the answer in the form \(\frac { \sqrt { m } + \sqrt { n } } { p }\), where \(m , n\) and \(p\) are integers.

1

1. The complex number \(\mathrm { a } + \mathrm { ib }\) is denoted by \(z\).
   1. Write down \(z ^ { * }\).
   2. Find \(\operatorname { Re } ( \mathrm { iz } )\).
2. The complex number \(w\) is given by \(w = \frac { 5 + \mathrm { i } \sqrt { 3 } } { 2 - \mathrm { i } \sqrt { 3 } }\).
   1. In this question you must show detailed reasoning. Express \(w\) in the form \(\mathrm { x } + \mathrm { iy }\).
   2. Convert \(w\) to modulus-argument form.

2 Two complex numbers are given by \(u = - 1 + \mathrm { i }\) and \(v = - 2 - \mathrm { i }\).

1. 1. Find \(\mathrm { u } - \mathrm { v }\) in the form \(\mathrm { a } + \mathrm { bi }\), where \(a\) and \(b\) are real.
   2. In this question you must show detailed reasoning. Find \(\frac { \mathrm { u } } { \mathrm { v } }\) in the form \(\mathrm { a } + \mathrm { bi }\), where \(a\) and \(b\) are real.
2. Express \(u\) in exact modulus-argument form.

1. The complex numbers \(z , v\) and \(w\) are related by the equation

$$z = \frac { v } { w }$$ Given that \(v = - 16 + 11 \mathrm { i }\) and \(w = 5 + 2 \mathrm { i }\), find \(z\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\).

1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

$$z _ { 1 } = - 4 + 4 i$$

1. Express \(\mathrm { z } _ { 1 }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r \in \mathbb { R } , r > 0\) and \(0 \leqslant \theta < 2 \pi\) $$z _ { 2 } = 3 \left( \cos \frac { 17 \pi } { 12 } + i \sin \frac { 17 \pi } { 12 } \right)$$
2. Determine in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are exact real numbers,
   1. \(\frac { Z _ { 1 } } { Z _ { 2 } }\)
   2. \(\left( z _ { 2 } \right) ^ { 4 }\)
3. Show on a single Argand diagram
   1. the complex numbers \(z _ { 1 } , z _ { 2 }\) and \(\frac { z _ { 1 } } { z _ { 2 } }\)
   2. the region defined by \(\left\{ z \in \mathbb { C } : \left| z - z _ { 1 } \right| < \left| z - z _ { 2 } \right| \right\}\)

1 In this question you must show detailed reasoning.
For the complex number \(z\) it is given that \(| z | = 2\) and \(\arg z = \frac { 1 } { 6 } \pi\).
Find the following in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are exact numbers.

1. \(z\)
2. \(z ^ { 2 }\)
3. \(\frac { z } { z ^ { * } }\)

2 In this question you must show detailed reasoning.
The complex number \(7 - 4 \mathrm { i }\) is denoted by \(z\).

1. Giving your answers in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are rational numbers, find the following.
   1. \(3 z - 4 z ^ { * }\)
   2. \(( z + 1 - 3 \mathrm { i } ) ^ { 2 }\)
   3. \(\frac { z + 1 } { z - 1 }\)
2. Express \(z\) in modulus-argument form giving the modulus exactly and the argument correct to 3 significant figures.
3. The complex number \(\omega\) is such that \(z \omega = \sqrt { 585 } ( \cos ( 0.5 ) + \mathrm { i } \sin ( 0.5 ) )\). Find the following.

11 The complex number \(w = ( \sqrt { 3 } - 1 ) + \mathrm { i } ( \sqrt { 3 } + 1 )\).

1. Determine, showing full working, the exact values of \(| w |\) and \(\arg w\).
   [0pt] [You may use the result that \(\tan \left( \frac { 5 } { 12 } \pi \right) = 2 + \sqrt { 3 }\).]
2. (a) Find, in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), the three roots, \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\), of the equation \(z ^ { 3 } = w\).
   (b) Determine \(z _ { 1 } z _ { 2 } z _ { 3 }\) in the form \(a + \mathrm { i } b\).
   (c) Mark the points representing \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) on a sketch of the Argand diagram. Show that they form an equilateral triangle, \(\Delta _ { 1 }\), and determine the side-length of \(\Delta _ { 1 }\).
   (d) The points representing \(k z _ { 1 } , k z _ { 2 }\) and \(k z _ { 3 }\) form \(\Delta _ { 2 }\), an equilateral triangle which is congruent to \(\Delta _ { 1 }\), and one of whose vertices lies on the positive real axis. Write down a suitable value for the complex constant \(k\).

Given that \(z = -2\sqrt{2} + 2\sqrt{2}i\) and \(w = 1 - i\sqrt{3}\), find

1. \(\left|\frac{z}{w}\right|\), [3]
2. \(\arg \left( \frac{z}{w} \right)\). [3]

1. On an Argand diagram, plot points \(A\), \(B\), \(C\) and \(D\) representing the complex numbers \(z\), \(w\), \(\left( \frac{z}{w} \right)\) and 4, respectively. [3]
2. Show that \(\angle AOC = \angle DOB\). [2]
3. Find the area of triangle \(AOC\). [2]

$$z = -4 + 6i.$$

1. Calculate \(\arg z\), giving your answer in radians to 3 decimal places. [2]

The complex number \(w\) is given by \(w = \frac{A}{2 - i}\), where \(A\) is a positive constant. Given that \(|w| = \sqrt{20}\),

1. find \(w\) in the form \(a + ib\), where \(a\) and \(b\) are constants, [4]
2. calculate \(\arg \frac{w}{z}\). [3]

The complex number \(z_1\) is \(1+ i\) and the complex number \(z_2\) has modulus 4 and argument \(\frac{\pi}{3}\).

1. Express \(z_2\) in the form \(a + bi\), giving \(a\) and \(b\) in exact form. [2]
2. Express \(\frac{z_2}{z_1}\) in the form \(c + di\), giving \(c\) and \(d\) in exact form. [2]

A complex number is defined by \(z = 3 + 4\mathrm{i}\).

1. Express \(z\) in the form \(z = re^{i\theta}\), where \(-\pi \leqslant \theta \leqslant \pi\). [3]
2. 1. Find the Cartesian coordinates of the vertices of the triangle formed by the cube roots of \(z\) when plotted in an Argand diagram. Give your answers correct to two decimal places.
   2. Write down the geometrical name of the triangle. [5]

When plotted on an Argand diagram, the four fourth roots of the complex number \(9 - 3\sqrt{3}i\) lie on a circle. Find the equation of this circle. [4]

You are given the complex number \(w = 2 + 2\sqrt{3}i\).

1. Express \(w\) in modulus-argument form. [3]
2. Indicate on an Argand diagram the set of points, \(z\), which satisfy both of the following inequalities. $$-\frac{\pi}{2} \leq \arg z \leq \frac{\pi}{3} \text{ and } |z| \leq 4$$ Mark \(w\) on your Argand diagram and find the greatest value of \(|z - w|\). [9]

In this question you must show detailed reasoning. The complex numbers \(z_1\) and \(z_2\) are given by \(z_1 = 2 - 3i\) and \(z_2 = a + 4i\) where \(a\) is a real number.

1. Express \(z_1\) in modulus-argument form, giving the modulus in exact form and the argument correct to 3 significant figures. [3]
2. Find \(z_1z_2\) in terms of \(a\), writing your answer in the form \(c + id\). [2]
3. The real and imaginary parts of a complex number on an Argand diagram are \(x\) and \(y\) respectively. Given that the point representing \(z_1z_2\) lies on the line \(y = x\), find the value of \(a\). [2]
4. Given instead that \(z_1z_2 = (z_1z_2)^*\) find the value of \(a\). [2]