Properties of Trigonometric Functions
Understanding zeros, signs, and periodicity of sine, cosine, tangent, and cotangent
1. Zeros of Sine
Recall that $\sin \alpha = y$, the $y$-coordinate of point $B$ on the unit circle. So $\sin \alpha = 0$ exactly when $y = 0$, meaning the terminal radius lands on the $x$-axis.
Zeros of sine: $\sin \alpha = 0$ when $\alpha = \pi n$, where $n$ is any integer.
$$\ldots,\; -2\pi,\; -\pi,\; 0,\; \pi,\; 2\pi,\; 3\pi,\; \ldots$$
$$\ldots,\; -2\pi,\; -\pi,\; 0,\; \pi,\; 2\pi,\; 3\pi,\; \ldots$$
On the unit circle, these correspond to the points $(1, 0)$ and $(-1, 0)$.