Answer
$$\frac{3}{2 \sqrt{2}}$$
Work Step by Step
We find:
$\kappa=\frac{\left|2\left(\frac{d r}{d \theta}\right)^{2}-r \frac{d^{2} r}{d \theta^{2}}+r^{2}\right|}{\left[r^{2}+\left(\frac{d r}{d \theta}\right)^{2}\right]^{3 / 2}}$
$-\sin \theta=\frac{d r}{d \theta}, \quad -\cos \theta=\frac{d^{2} r}{d \theta^{2}}$
$\kappa(\theta)=\frac{\left|1+2 \cos \theta+\cos ^{2} \theta+2(-\sin \theta)^{2}-(1+\cos \theta)(-\cos \theta)\right|}{\left[\cos ^{2} \theta+2 \cos \theta+(-\cos \theta)^{2}+1\right]^{3 / 2}}$
$=\frac{3}{2 \sqrt{2}(\cos \theta+1)^{1 / 2}}$
$\frac{3}{2 \sqrt{2}}=\kappa(\pi / 2)$
