在同一坐标轴上画出在 $0^\circ \le x \le 360^\circ$ 范围内的 $y = -\sin x + 1$ 和 $y = 3\cos 2x$ 的图像草图。 由此，确定方程 $-\sin x + 1 = 3\cos 2x$ 在 $0^\circ \le x \le 360^\circ$ 范围内的解的个数。

1. Determine the key features and points for $y = -\sin x + 1$.

For $y = -\sin x + 1$:

• Amplitude is $1$.
• Period is $360^\circ$.
• The graph is a standard sine curve reflected in the $x$-axis and translated upwards by $1$ unit.
• Key points: $(0^\circ, 1)$, $(90^\circ, 0)$, $(180^\circ, 1)$, $(270^\circ, 2)$, $(360^\circ, 1)$.

2. Determine the key features and points for $y = 3\cos 2x$.

For $y = 3\cos 2x$:

• Amplitude is $3$.
• Period is $\frac{360^\circ}{2} = 180^\circ$.
• The graph completes two full cycles between $0^\circ$ and $360^\circ$.
• Key points for the first cycle: $(0^\circ, 3)$, $(45^\circ, 0)$, $(90^\circ, -3)$, $(135^\circ, 0)$, $(180^\circ, 3)$.
• Key points for the second cycle: $(225^\circ, 0)$, $(270^\circ, -3)$, $(315^\circ, 0)$, $(360^\circ, 3)$.

1. Determine the number of solutions by counting the points of intersection of the two graphs

The solutions to the equation $-\sin x + 1 = 3\cos 2x$ correspond to the points of intersection of the graphs $y = -\sin x + 1$ and $y = 3\cos 2x$.
From the sketch in part (a), the two graphs intersect at $4$ distinct points in the interval $0^\circ \le x \le 360^\circ$.