In translational motion, the work $W$ done by a constant force $\vec{F}$ acting at an angle $\phi$ to a displacement $d$ is defined as $W = |\vec{F}| d \cos \phi$.

Rotational motion follows a strictly analogous pattern.

If an object rotates about a fixed axis while under the influence of a constant net torque $\tau$, the work done by that torque depends on the angular displacement $\Delta \theta$ of the point of application of the force causing the torque.

Derivation

To see this relationship, consider a force $\vec{F}$ applied at an angle of $\phi$ below the horizontal to the edge of a wheel of radius $r$.

The magnitude of the torque produced is $\tau = r |\vec{F}|\sin\phi$. When the wheel rotates through an angle $\Delta \theta$, the point where the force is applied moves through an arc length $d = r \Delta \theta$.

The displacement $d$ is along the tangent direction (downward at this point), which makes an angle of $(90^\circ - \phi)$ with the force $\vec{F}$.