Key Trig Values 175D2D

1. The problem asks to fill in the sine and cosine values for the angles $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$, and $90^\circ$ without looking. 2. Recall the key trigonometric values for sine and cosine at these special angles: - $\sin 0^\circ = 0$ - $\cos 0^\circ = 1$ - $\sin 30^\circ = \frac{1}{2}$ - $\cos 30^\circ = \frac{\sqrt{3}}{2}$ - $\sin 45^\circ = \frac{\sqrt{2}}{2}$ - $\cos 45^\circ = \frac{\sqrt{2}}{2}$ - $\sin 60^\circ = \frac{\sqrt{3}}{2}$ - $\cos 60^\circ = \frac{1}{2}$ - $\sin 90^\circ = 1$ - $\cos 90^\circ = 0$ 3. These values come from the unit circle and special right triangles (30-60-90 and 45-45-90 triangles). 4. Filling the table: | Angle | $\sin$ | $\cos$ | |-------|--------|--------| | $0^\circ$ | $0$ | $1$ | | $30^\circ$ | $\frac{1}{2}$ | $\frac{\sqrt{3}}{2}$ | | $45^\circ$ | $\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{2}}{2}$ | | $60^\circ$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{2}$ | | $90^\circ$ | $1$ | $0$ | 5. These are the exact values, not decimal approximations, which are preferred for precision in trigonometry.