A complex number is defined by \(z = 3 + 4 \mathrm { i }\).
Express \(z\) in the form \(z = r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(- \pi \leqslant \theta \leqslant \pi\).
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.
Write down the geometrical name of the triangle.
(a) Show that \(3 \sin x + 4 \cos x - 2\) can be written as \(\frac { 6 t + 2 - 6 t ^ { 2 } } { 1 + t ^ { 2 } }\), where \(t = \tan \left( \frac { x } { 2 } \right)\).
Hence, find the general solution of the equation \(3 \sin x + 4 \cos x - 2 = 3\).
(a) Determine whether or not the following set of equations
$$\left( \begin{array} { r r r }
2 & - 7 & 2
0 & 3 & - 2
- 7 & 8 & 4
\end{array} \right) \left( \begin{array} { l }
x
y
z
\end{array} \right) = \left( \begin{array} { l }
a
b
c
\end{array} \right)$$
has a unique solution, where \(a , b , c\) are constants.
Solve the set of equations
$$\begin{aligned}
x + 8 y - 6 z & = 5
2 x + 4 y + 6 z & = - 3
- 5 x - 4 y + 9 z & = - 7
\end{aligned}$$
Show all your working.