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