Sketch on an Argand diagram the points represented by w⁰ , w¹ , w² and w³.

For this value of $\theta$, plot the approximate position of $z_{\theta }$ on the Argand diagram.

$\theta =\mathrm{\pi }$.

Find this value of $\theta$.

Hence write down the maximum voltage in the circuit.

draw $w$, $w^2$, $w^3$, and $w^4$ on the following Argand diagram.

Find the value of ${z_1}^3$.

$\theta =\frac{3\mathrm{\pi }}{2}$.

Describe these two transformations and give the order in which they are applied.

An equilateral triangle is to be drawn on the Argand plane with one of the vertices at the point corresponding to $2+5\text{i}$ and all the vertices equidistant from $0$.

Find the points that correspond to the other two vertices. Give your answers in Cartesian form.

Find an expression for $V_T$ in the form $A \cos (Bt+C)$, where $A, B$ and $C$ are real constants.

Find the maximum value of $I_{\text{T}}$.

$\theta =\frac{\mathrm{\pi }}{2}$.

Hence find $\text{Re}\left(w_1+w_2\right)$ in the form $A \cos \left(x-a\right)$, where $A>0$ and $0<a\le \frac{\mathrm{\pi }}{2}$.

Find $w_1+w_2$ in the form ${\text{e}}^{\text{i}x}\left(c+\text{i}d\right)$.

Show the points represented by $z$ and $z-2a$ on the following Argand diagram.

Express w² and w³ in modulus-argument form.

Hence find the cube roots of $z$ in modulus-argument form.

By expressing $z_1$ and $z_2$ in modulus-argument form write down the argument of $w$.

Let $z=\frac{w}{2-\text{i}}$.

Find the value of $a$ for which successive powers of $z$ lie on a circle.

Find the modulus and argument of $z$ in terms of $\theta$. Express each answer in its simplest form.

In the following Argand diagram the point A represents the complex number $-1+4\text{i}$ and the point B represents the complex number $-3+0\text{i}$. The shape of ABCD is a square. Determine the complex numbers represented by the points C and D.

Find the smallest positive integer value of $n$, such that $w^n$ is a real number.

Show that the area of the quadrilateral Q is $\frac{21\sqrt{3}}{2}$.

By expressing $z_1$ and $z_2$ in modulus-argument form write down the modulus of $w$;

Solve $2\sin (x+{60}^{\circ })=\cos (x+{30}^{\circ }),0^{\circ }⩽x⩽{180}^{\circ }$.

$z=1+\text{i}$.

$z=2\text{i}$.

Find, in hours, the shortest time from sunrise to sunset at point $\text{A}$ that is predicted by this model.

Let $z=2\left(\text{cos}\frac{\pi }{n}+\text{i}\text{sin}\frac{\pi }{n}\right),n\in {\mathbb{Z}}^{+}$. The points represented on an Argand diagram by $z^0,z^1,z^2,\dots ,z^n$ form the vertices of a polygon $P_n$.

Show that the area of the polygon $P_n$ can be expressed in the form $a\left(b^n-1\right)\text{sin}\frac{\pi }{n}$, where $a,b\in \mathbb{R}$.

Hence or otherwise find an equation for $L$ in the form $L\left(t\right)=p \sin (qt+r)+d$, where $p, q, r, d\in \mathrm{ℝ}$.

Show that $\sin {105}^{\circ }+\cos {105}^{\circ }=\frac{1}{\sqrt{2}}$.

Write down $2+5\text{i}$ in exponential form.

Write down $z_1$ and $z_2$ in exponential form, with a constant modulus.

Find the least value of $n$ such that $z_1{z}_2\in {\mathrm{ℝ}}^{+}$.

Hence, or otherwise, find the value of $z$ when $w=2-\text{i}$.

Find the value of ${\left(\frac{z_1}{z_2}\right)}^4$ for $n=2$.

Plot the position of $z$ on an Argand Diagram.

Express $z$ in the form $z=a{\text{e}}^{\text{i}b}$, where $a, b\in \mathrm{ℝ}$, giving the exact value of $a$ and the exact value of $b$.

Find the phase shift of $I_{\text{T}}$.