Argument calculations and identities

7 The complex number \(2 + \mathrm { i }\) is denoted by \(u\). Its complex conjugate is denoted by \(u ^ { * }\).

1. Show, on a sketch of an Argand diagram with origin \(O\), the points \(A , B\) and \(C\) representing the complex numbers \(u , u ^ { * }\) and \(u + u ^ { * }\) respectively. Describe in geometrical terms the relationship between the four points \(O , A , B\) and \(C\).
2. Express \(\frac { u } { u ^ { * } }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
3. By considering the argument of \(\frac { u } { u ^ { * } }\), or otherwise, prove that $$\tan ^ { - 1 } \left( \frac { 4 } { 3 } \right) = 2 \tan ^ { - 1 } \left( \frac { 1 } { 2 } \right) .$$

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.

5. \begin{figure}[h] \end{figure} The complex numbers \(z _ { 1 } = - 2 , z _ { 2 } = - 1 + 2 \mathrm { i }\) and \(z _ { 3 } = 1 + \mathrm { i }\) are plotted in Figure 1, on an Argand diagram for the complex plane with \(z = x + \mathrm { i } y\)

1. Explain why \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) cannot all be roots of a quartic polynomial equation with real coefficients.
2. Show that \(\arg \left( \frac { z _ { 2 } - z _ { 1 } } { z _ { 3 } - z _ { 1 } } \right) = \frac { \pi } { 4 }\)
3. Hence show that \(\arctan ( 2 ) - \arctan \left( \frac { 1 } { 3 } \right) = \frac { \pi } { 4 }\) A copy of Figure 1, labelled Diagram 1, is given on page 12.
4. Shade, on Diagram 1, the set of points of the complex plane that satisfy the inequality $$| z + 2 | \leqslant | z - 1 - \mathrm { i } |$$
   \includegraphics[max width=\textwidth, alt={}]{9312b91c-bca7-4427-a1f7-cb03065ee5e5-12_479_524_296_776}
   \section*{Diagram 1}

1. A student was asked to answer the following:

For the complex numbers \(z _ { 1 } = 3 - 3 \mathrm { i }\) and \(z _ { 2 } = \sqrt { 3 } + \mathrm { i }\), find the value of \(\arg \left( \frac { z _ { 1 } } { z _ { 2 } } \right)\) The student's attempt is shown below. \includegraphics[max width=\textwidth, alt={}, center]{33292670-3ad0-4125-a3bb-e4b7b21ed5f4-02_798_1109_534_338} The student made errors in line 1 and line 3
Correct the error that the student made in

1. 1. line 1
   2. line 3
2. Write down the correct value of \(\arg \left( \frac { z _ { 1 } } { z _ { 2 } } \right)\)

2 Given that arg \(( a + b \mathrm { i } ) = \varphi\), where \(a\) and \(b\) are positive real numbers and \(0 < \varphi < \frac { \pi } { 2 }\), three of the following four statements are correct. Which statement is not correct? Tick \(( \checkmark )\) one box. $$\begin{aligned} & \arg ( - a - b \mathrm { i } ) = \pi - \varphi \\ & \arg ( a - b \mathrm { i } ) = - \varphi \\ & \arg ( b + a \mathrm { i } ) = \frac { \pi } { 2 } - \varphi \\ & \arg ( b - a \mathrm { i } ) = \varphi - \frac { \pi } { 2 } \end{aligned}$$

2 Find arg ( \(- 4 - 7 \mathrm { i }\) ) to the nearest degree.
Circle your answer.
[0pt] [1 mark] \(- 120 ^ { \circ }\) \(- 60 ^ { \circ }\) \(30 ^ { \circ }\) \(60 ^ { \circ }\)