Step 1: Spot a shortcut using the 30 degree angle.
Whenever the semi vertical angle of a cone is $30^\circ$, the radius and the slant height are tied together in a very simple way, because $\sin 30^\circ = \tfrac12$ always gives $l = 2r$, no matter what the actual radius is.
Step 2: Apply this to our cone.
Here $r = 7$ cm, so the slant height is simply \[ l = 2r = 2 \times 7 = 14 \text{ cm} \]
Step 3: Use the curved surface area formula.
The curved surface area of a cone is $\text{CSA} = \pi r l$. Putting in the values, \[ \text{CSA} = \frac{22}{7} \times 7 \times 14 = 22 \times 14 = 308 \text{ cm}^2 \]
Step 4: State the result.
So the curved surface area works out to 308 sq cm, the same value we would reach by first solving for l using the sine ratio.
\[ \boxed{308 \text{ cm}^2} \]