Answer
$\color{blue}{5\sqrt{2}\ \text{cis}\ 45^\circ,\ 5\sqrt{2}\ \text{cis}\ \pi/4}$
Work Step by Step
$z=5+5i = x+iy \implies x=5, y=5$
$\Huge\cdot$ modulus: $\quad r = \sqrt{x^2+y^2} = \sqrt{5^2+5^2}= \sqrt{2(25)} = 5\sqrt{2}$
$\Huge\cdot$ argument: $\quad \tan\theta = y/x=5/5=1 \implies \theta = 45^\circ \equiv \pi/4$ (smallest positive real angle $\theta$ from $+x$-axis to graph of $z$)
$\begin{array}{|c|c|c|} \hline
\text{Standard} & \text{Trigonometric} & \text{Trigonometric} \\
\text{Form} & \text{Form (deg)} & \text{Form (rad)} \\ \hline
5+5i & 5\sqrt{2}\ \text{cis}\ 45^\circ & 5\sqrt{2}\ \text{cis}\ \pi/4 \\ \hline
\end{array}$
