The points P and Q are represented by the complex numbers p and q respectively, where $p=\sqrt{2}+\sqrt{2}i\,,\ \arg(q)=\dfrac{2}{3}\pi\text{ and }|q|=2.$

1. Calculate the modulus of $p$

$|p| = \sqrt{(\sqrt{2})^2 + (\sqrt{2})^2} = \sqrt{2 + 2} = \sqrt{4} = 2$

2. Calculate the principal argument of $p$

Since the real and imaginary parts are both positive, $p$ lies in the first quadrant.
$\arg(p) = \tan^{-1}\left(\dfrac{\sqrt{2}}{\sqrt{2}}\right) = \tan^{-1}(1) = \dfrac{\pi}{4}$

1. Identify the locations of P and Q based on their modulus and argument

From the given information and part (a):
$|p| = 2$, $\arg(p) = \frac{\pi}{4}$
$|q| = 2$, $\arg(q) = \frac{2\pi}{3}$

Both points have the same modulus (distance of 2 from the origin). P lies in the first quadrant (since $0 < \frac{\pi}{4} < \frac{\pi}{2}$) and Q lies in the second quadrant (since $\frac{\pi}{2} < \frac{2\pi}{3} < \pi$).

2. Describe the sketch of the Argand diagram

• Draw the horizontal axis and label it $\text{Re}$. Draw the vertical axis and label it $\text{Im}$.
• Mark the origin as $O$.
• Plot point P in the 1st quadrant. Draw the line segment $OP$, indicating its length is $2$ and the angle it makes with the positive $\text{Re}$ axis is $\frac{\pi}{4}$.
• Plot point Q in the 2nd quadrant. Draw the line segment $OQ$, indicating its length is $2$ (visually the same length as $OP$) and the angle it makes with the positive $\text{Re}$ axis is $\frac{2\pi}{3}$.

1. Deduce the real part of $q$ using its horizontal component on the Argand diagram

From the Argand diagram, the point Q has a distance of $2$ from the origin and makes an angle of $\frac{2}{3}\pi$ with the positive real axis (putting it in the second quadrant). The horizontal component gives the real part:
$\text{Re}(q) = 2\cos\left(\frac{2}{3}\pi\right) = 2\left(-\frac{1}{2}\right) = -1$

2. Deduce the imaginary part of $q$ using its vertical component on the Argand diagram

Similarly, the vertical component gives the imaginary part:
$\text{Im}(q) = 2\sin\left(\frac{2}{3}\pi\right) = 2\left(\frac{\sqrt{3}}{2}\right) = \sqrt{3}$

1. Identify the shape formed by complex addition.

Since the point $R$ is represented by $p+q$, the addition of the complex numbers means that $OPRQ$ is a parallelogram.

2. Use the lengths of the adjacent sides to deduce the specific shape.

From part (a), $|p| = 2$, which gives the length $OP = 2$.
We are given $|q| = 2$, giving the length $OQ = 2$.
Because the adjacent sides of the parallelogram $OPRQ$ are equal in length ($OP = OQ = 2$), the quadrilateral is a rhombus.

1. Express $p+q$ in Cartesian form using the values found in earlier parts.

From earlier parts, $p = \sqrt{2} + \sqrt{2}i$ and $q = -1 + \sqrt{3}i$.
Therefore, the sum is:
$p+q = (\sqrt{2}-1) + (\sqrt{2}+\sqrt{3})i$

2. Determine $\arg(p+q)$ geometrically using the properties of the quadrilateral OPRQ.

From part (d), quadrilateral OPRQ is a rhombus.
In an Argand diagram, the diagonal OR represents the complex number $p+q$. Since the diagonal of a rhombus bisects the interior angle, the argument of $p+q$ is the exact average of the arguments of $p$ and $q$:
$\arg(p+q) = \frac{\arg(p) + \arg(q)}{2}$
$\arg(p+q) = \frac{\frac{\pi}{4} + \frac{2\pi}{3}}{2} = \frac{\frac{3\pi}{12} + \frac{8\pi}{12}}{2} = \frac{11\pi}{24}$

3. Equate the tangent of $\arg(p+q)$ to the ratio of its imaginary and real parts.

For any complex number $z = x + iy$, $\tan(\arg(z)) = \frac{y}{x}$.
Using the Cartesian form of $p+q$ from Step 1:
$\tan(\arg(p+q)) = \frac{\text{Im}(p+q)}{\text{Re}(p+q)}$
$\tan\left(\frac{11\pi}{24}\right) = \frac{\sqrt{2}+\sqrt{3}}{\sqrt{2}-1}$

4. Rationalise the denominator to obtain the final shown result.

To show the required result, rationalise the denominator by multiplying the numerator and denominator by the conjugate $\sqrt{2}+1$:
$\tan\left(\frac{11\pi}{24}\right) = \frac{\sqrt{2}+\sqrt{3}}{\sqrt{2}-1} \times \frac{\sqrt{2}+1}{\sqrt{2}+1}$
$= \frac{(\sqrt{2}+\sqrt{3})(\sqrt{2}+1)}{(\sqrt{2})^2 - 1^2}$
$= \frac{2 + \sqrt{2} + \sqrt{6} + \sqrt{3}}{2 - 1}$
$= \sqrt{6} + \sqrt{3} + \sqrt{2} + 2 \quad \text{(shown)}$