1. The problem is to find the integral of $\sin \theta$ with respect to $\theta$. 2. The formula for the integral of $\sin x$ is: $$\int \sin x \, dx = -\cos x + C$$ where $C$ is the constant of integration. 3. Applying this formula to our problem: $$\int \sin \theta \, d\theta = -\cos \theta + C$$ 4. This means the antiderivative of $\sin \theta$ is $-\cos \theta$ plus a constant. 5. So, the final answer is: $$\boxed{-\cos \theta + C}$$