Introduction to trigonometry

Aaron knows that $\tan{\theta} = \dfrac{\sin{\theta}}{\cos{\theta}}$ . Therefore, $\tan{\theta} = \sin{\theta} \quad \forall \quad \theta$ such that $\cos{\theta} = 1$ , or $\sin{\theta} = 0$ . He notes that this happens when $\theta = 0$ . Since $\left | \cos{\theta} \right | \leq 1 \quad \forall \quad \theta$ , we see that the magnitude of $\tan{\theta}$ is at least that of $\sin{\theta}$ . As shown earlier, we know that $\sin{0} = \tan{0} = 0$ , however this also happens for $\sin{\theta} = \tan{\theta} = 0$ when $\theta = n\pi, \ n \in \mathbb{Z}$ . Therefore, Aaron is incorrect because he didn't account for the fact that $\sin{\theta}$ is a periodic function and has an infinite number of zeros. A correct statement would have been $\left | \sin{\theta} \right | \leq \left | \tan{\theta} \right | \quad \forall \quad \theta$ .