If we equate the arclengths and account for the opposite rotation directions, we obtain:

${\mathrm{R}}_{\mathrm{large}} {\mathrm{\varphi }}_{\mathrm{large}}=-{\mathrm{R}}_{\mathrm{small} }{\mathrm{\varphi }}_{\mathrm{small}}$

This equation expresses the relationship between rotation angles for the two gears. Solving for ${\mathrm{\varphi }}_{\mathrm{large}}$:

${\mathrm{\varphi }}_{\mathrm{large}}=-\left(\frac{{\mathrm{R}}_{\mathrm{small}}}{{\mathrm{R}}_{\mathrm{large}}}\right){\mathrm{\varphi }}_{\mathrm{small}}=-\left(\frac{{\mathrm{D}}_{\mathrm{small}}}{{\mathrm{D}}_{\mathrm{large}}}\right){\mathrm{\varphi }}_{\mathrm{small}}=-\left(\frac{{\mathrm{N}}_{\mathrm{small}}}{{\mathrm{N}}_{\mathrm{large}}}\right){\mathrm{\varphi }}_{\mathrm{small}}$