Modulus-argument form conversion

7. $$z = - 24 - 7 i$$

1. Show \(z\) on an Argand diagram.
2. Calculate \(\arg z\), giving your answer in radians to 2 decimal places. It is given that $$w = a + b \mathrm { i } , \quad a \in \mathbb { R } , b \in \mathbb { R }$$ Given also that \(| w | = 4\) and \(\arg w = \frac { 5 \pi } { 6 }\),
3. find the values of \(a\) and \(b\),
4. find the value of \(| z w |\).

3. $$z _ { 1 } = \frac { 1 } { 2 } ( 1 + \mathrm { i } \sqrt { } 3 ) , z _ { 2 } = - \sqrt { } 3 + \mathrm { i }$$

1. Express \(z _ { 1 }\) and \(z _ { 2 }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\) giving exact values of \(r\) and \(\theta\).
   (4)
2. Find \(\left| z _ { 1 } z _ { 2 } \right|\).
3. Show and label \(z _ { 1 }\) and \(z _ { 2 }\) on a single Argand diagram.
   (2)

1 The complex number \(a + \mathrm { i } b\) is denoted by \(z\). Given that \(| z | = 4\) and \(\arg z = \frac { 1 } { 3 } \pi\), find \(a\) and \(b\).

1 The complex number \(a + 5 \mathrm { i }\), where \(a\) is positive, is denoted by \(z\). Given that \(| z | = 13\), find the value of \(a\) and hence find \(\arg z\).

8 Abdoallah wants to write the complex number \(- 1 + \mathrm { i } \sqrt { 3 }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\) where \(r \geq 0\) and \(- \pi < \theta \leq \pi\) Here is his method: $$\begin{array} { r l r l } r & = \sqrt { ( - 1 ) ^ { 2 } + ( \sqrt { 3 } ) ^ { 2 } } & & \tan \theta = \frac { \sqrt { 3 } } { - 1 } \\ & = \sqrt { 1 + 3 } & & \Rightarrow \\ & = \sqrt { 4 } & & \tan \theta = - \sqrt { 3 } \\ & = 2 & & \theta = \tan ^ { - 1 } ( - \sqrt { 3 } ) \\ & & \theta = - \frac { \pi } { 3 } \\ & - 1 + i \sqrt { 3 } = 2 \left( \cos \left( - \frac { \pi } { 3 } \right) + i \sin \left( - \frac { \pi } { 3 } \right) \right) \end{array}$$ There is an error in Abdoallah's method. 8

1. Show that Abdoallah's answer is wrong by writing $$2 \left( \cos \left( - \frac { \pi } { 3 } \right) + i \sin \left( - \frac { \pi } { 3 } \right) \right)$$ in the form \(x + \mathrm { i } y\) Simplify your answer.
   8
2. Explain the error in Abdoallah's method.
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3. Express \(- 1 + \mathrm { i } \sqrt { 3 }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\) 8
4. Write down the complex conjugate of \(- 1 + i \sqrt { 3 }\)

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

1. 1. Express \(z\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(- \pi \leqslant \theta \leqslant \pi\).
   2. Express \(\bar { z }\), the complex conjugate of \(z\), in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\). Another complex number is defined as \(w = \sqrt { 5 } ( \cos 2 \cdot 68 + \mathrm { i } \sin 2 \cdot 68 )\).
2. Express \(z w\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\).

1 The complex number \(z _ { 1 }\) is \(1 + \mathrm { i }\) and the complex number \(z _ { 2 }\) has modulus 4 and argument \(\frac { \pi } { 3 }\).

1. Express \(z _ { 2 }\) in the form \(a + b \mathrm { i }\), giving \(a\) and \(b\) in exact form.
2. Express \(\frac { z _ { 2 } } { z _ { 1 } }\) in the form \(c + d i\), giving \(c\) and \(d\) in exact form.
3. Describe fully the transformation represented by the matrix \(\left( \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right)\).
4. A triangle of area 5 square units undergoes the transformation represented by the matrix \(\left( \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right)\). Explaining your reasoning, find the area of the image of the triangle following this transformation.

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.

3 In this question you must show detailed reasoning.
The complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are given by \(z _ { 1 } = - 2 + 2 i\) and \(z _ { 2 } = 2 \left( \cos \frac { 1 } { 6 } \pi + i \sin \frac { 1 } { 6 } \pi \right)\).

1. Find the modulus and argument of \(z _ { 1 }\).
2. Hence express \(\frac { z _ { 1 } } { z _ { 2 } }\) in exact modulus-argument form.

3. The complex numbers \(z\) and \(w\) are represented by the points \(Z\) and \(W\) in an Argand diagram. The complex number \(z\) is such that \(| z | = 6\) and \(\arg z = \frac { \pi } { 3 }\).
The point \(W\) is a \(90 ^ { \circ }\) clockwise rotation, about the origin, of the point \(Z\) in the Argand diagram.

1. Express \(z\) and \(w\) in the form \(x + \mathrm { i } y\).
2. Find the complex number \(\frac { z } { w }\).

1. Given that

$$\begin{aligned} z _ { 1 } & = 2 + 3 \\ \left| z _ { 1 } z _ { 2 } \right| & = 39 \sqrt { 2 } \\ \arg \left( z _ { 1 } z _ { 2 } \right) & = \frac { \pi } { 4 } \end{aligned}$$ where \(z _ { 1 }\) and \(z _ { 2 }\) are complex numbers,

1. write \(z _ { 1 }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\) Give the exact value of \(r\) and give the value of \(\theta\) in radians to 4 significant figures.
2. Find \(z _ { 2 }\) giving your answer in the form \(a + \mathrm { i } b\) where \(a\) and \(b\) are integers.

3 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. \(( z + 1 - 3 i ) ^ { 2 }\)
   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.
   • \(| \omega |\)
   • \(\arg ( \omega )\), giving your answer correct to 3 significant figures

1

1. The complex number 3-4i is denoted by \(z _ { 1 }\). Write \(z _ { 1 }\) in modulus-argument form, giving your angle in radians to 3 significant figures.
2. The complex number \(z _ { 2 }\) has modulus 6 and argument - 2.5 radians. Express \(z _ { 1 } z _ { 2 }\) in modulus-argument form with the angle in radians correct to 3 significant figures.

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. \(\frac { z } { z ^ { * } }\)

15. Given that \(z = 2 - 2 \mathrm { i }\) and \(w = - \sqrt { 3 } + \mathrm { i }\),

1. find the modulus and argument of \(w z ^ { 2 }\).
   (6)
2. Show on an Argand diagram the points \(A , B\) and \(C\) which represent \(z , w\) and \(w z ^ { 2 }\) respectively, and determine the size of angle \(B O C\).

1 Express the complex number \(1 - \mathrm { i } \sqrt { 3 }\) in modulus-argument form.
Tick ( \(\checkmark\) ) one box. $$\begin{array} { l l } 2 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right) & \square \\ 2 \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right) & \square \\ 2 \left( \cos \left( - \frac { \pi } { 3 } \right) + i \sin \left( - \frac { \pi } { 3 } \right) \right) & \square \\ 2 \left( \cos \left( - \frac { 2 \pi } { 3 } \right) + i \sin \left( - \frac { 2 \pi } { 3 } \right) \right) \end{array}$$

1 The complex number \(\omega\) is shown below on the Argand diagram. \includegraphics[max width=\textwidth, alt={}, center]{f7e7c21b-6e72-4c20-92fc-ba0336a11136-02_597_650_632_689} Which of the following complex numbers could be \(\omega\) ?
Tick ( \(\checkmark\) ) one box. $$\begin{aligned} & \cos ( - 2 ) + i \sin ( - 2 ) \\ & \cos ( - 1 ) + i \sin ( - 1 ) \\ & \cos ( 1 ) + i \sin ( 1 ) \\ & \cos ( 2 ) + i \sin ( 2 ) \end{aligned}$$ □
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1 You are given that \(z = 3 - 4 \mathrm { i }\).

1. Find
   • \(| z |\),
   • \(\arg ( z )\),
   • \(Z ^ { * }\).
   On an Argand diagram the complex number \(w\) is represented by the point \(A\) and \(w ^ { * }\) is represented by the point \(B\).
2. Describe the geometrical relationship between the points \(A\) and \(B\).