1. The problem is to understand how to know the special angles in trigonometry. 2. Special angles are commonly used angles where the sine, cosine, and tangent values are well-known and easy to remember. These angles are typically $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$, and $90^\circ$ (or in radians: $0$, $\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$, $\frac{\pi}{2}$). 3. The sine, cosine, and tangent values for these angles can be derived from special right triangles: the 45-45-90 triangle and the 30-60-90 triangle. 4. For example, in a 45-45-90 triangle, the sides are in the ratio $1:1:\sqrt{2}$, so: $$\sin 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}, \quad \cos 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}, \quad \tan 45^\circ = 1$$ 5. In a 30-60-90 triangle, the sides are in the ratio $1:\sqrt{3}:2$, so: $$\sin 30^\circ = \frac{1}{2}, \quad \cos 30^\circ = \frac{\sqrt{3}}{2}, \quad \tan 30^\circ = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$$ $$\sin 60^\circ = \frac{\sqrt{3}}{2}, \quad \cos 60^\circ = \frac{1}{2}, \quad \tan 60^\circ = \sqrt{3}$$ 6. Memorizing these values or understanding the triangles helps you quickly know the special angles and their trigonometric values. 7. Remember that $\sin 0^\circ = 0$, $\cos 0^\circ = 1$, and $\tan 0^\circ = 0$, and at $90^\circ$, $\sin 90^\circ = 1$, $\cos 90^\circ = 0$, and $\tan 90^\circ$ is undefined. This knowledge is fundamental in trigonometry and helps solve many problems efficiently.