Electrical field outside spherical shell

Homework Statement

An insulated spherical shell with its center in the origin and radius R has a surface charge density of $$\sigma(\theta)=\sigma_0\sin(\theta)$$ when $$\theta$$ is the angle from the z axis. Calculate the electrical field outside the shell at point z=R.

Homework Equations

Gauss's Law
$$\int\int{\vec{E}\cdot\vec{dS}}=4\pi{Q_{in}}$$

The Attempt at a Solution

I tried first calculating the overall charge of the spherical shell:
$$Q=R^2\int_0^{2\pi}\sigma_0\sin^2\theta{d\theta}\int_0^\pi{d\phi}=\frac{R^2\sigma_0}{2}\int_0^{2\pi}(1-\cos{2\theta})d\theta\cdot\pi=\frac{1}{2}\pi{R^2}\sigma_0\cdot{2\pi}=\pi^2R^2\sigma_0$$
However, I don't see how Gauss's Law can help me find the electrical field at that specific point.

Perhaps I should somehow calculate $$\int{\frac{dq}{r^2}}$$ ?