Modulus of integral

Relevant wiki: Euler's Formula

$\begin{aligned} z & = \int_\frac \pi 2^\pi \cos \theta {\color{#3D99F6}e^{i\theta}} d\theta & \small \color{#3D99F6} \text{By Euler's formula }e^{ix} = \cos x + i \sin x \\ & = \int_\frac \pi 2^\pi \cos \theta {\color{#3D99F6}(\cos \theta + i \sin \theta)} d\theta \\ & = \int_\frac \pi 2^\pi \left(\cos^2 \theta + i \sin \theta \cos \theta \right) d\theta \\ & = \int_\frac \pi 2^\pi \frac 12 \left(1 + \cos 2 \theta + i \sin 2 \theta \right) d\theta \\ & = \frac 12 \left[ \theta + \frac {\sin 2\theta}2 - i \frac {\cos 2\theta}2\right]_\frac \pi 2^\pi \\ & = \frac \pi 4 - \frac i2 \end{aligned}$

$\begin{aligned} \implies |z| & = \left|\frac \pi 4 - \frac i2\right| = \sqrt{\frac {\pi^2}{16}+\frac 14} \approx \boxed{0.931} \end{aligned}$