函数 $y = 5 \sin 3x + c$ 的图像经过点 $(50^\circ, \frac{1}{2})$，其中 $c$ 为常数。

1. Substitute the coordinates of the given point into the equation

Substitute $x = 50^\circ$ and $y = \frac{1}{2}$ into $y = 5 \sin 3x + c$:
$\frac{1}{2} = 5 \sin(3 \times 50^\circ) + c$
$\frac{1}{2} = 5 \sin(150^\circ) + c$

2. Evaluate the trigonometric function and solve for $c$

Since $\sin(150^\circ) = \frac{1}{2}$:
$\frac{1}{2} = 5\left(\frac{1}{2}\right) + c$
$\frac{1}{2} = \frac{5}{2} + c$
$c = \frac{1}{2} - \frac{5}{2} = -\frac{4}{2} = -2$

1. Identify the amplitude, period, and principal axis of the function

From (i), $c = -2$, so the equation is $y = 5 \sin 3x - 2$.
Amplitude $= 5$
Period $= \frac{360^\circ}{3} = 120^\circ$
Principal axis is $y = -2$.

2. Determine the maximum and minimum values

Maximum value $= -2 + 5 = 3$
Minimum value $= -2 - 5 = -7$

3. Determine key coordinates and sketch the graph for $0^\circ \le x \le 180^\circ$

The domain $0^\circ \le x \le 180^\circ$ covers $1.5$ periods.
Key points:

• $x = 0^\circ, y = -2$
• $x = 30^\circ, y = 3$ (Maximum)
• $x = 60^\circ, y = -2$
• $x = 90^\circ, y = -7$ (Minimum)
• $x = 120^\circ, y = -2$
• $x = 150^\circ, y = 3$ (Maximum)
• $x = 180^\circ, y = -2$

Sketch a smooth sine curve passing through these points.