Step 1: Write the minimum teeth formula.
For a \(20^\circ\) full depth involute pinion meshing with a rack, the minimum number of teeth to avoid interference is:
\[ T_{\min} = \frac{2a_w}{\sin^2\phi}, \qquad a_w = 1,\ \phi = 20^\circ \]
Step 2: Get \(\sin^2 20^\circ\) using the double angle identity.
Instead of squaring \( \sin 20^\circ \) directly, use \( \sin^2\theta = \dfrac{1-\cos 2\theta}{2} \):
\[ \sin^2(20^\circ) = \frac{1-\cos(40^\circ)}{2} = \frac{1-0.766}{2} \approx 0.117 \]
Step 3: Substitute and round up to a whole number of teeth.
\[ T_{\min} = \frac{2}{0.117} \approx 17.1 \]
Since a fraction of a tooth is not physically possible and interference must be fully avoided, round up:
\[ \boxed{T_{\min} = 18} \]
This matches option (C).