I think the magnetic field is assumed to be uniform B=B[sub]0[/sub].
The magnetic flux pass through the coil depends on the angle $\theta=\omega t$,
assume the coil rotate at constant angular velocity $\omega$.
i.e. $\Psi=\int \vec{B}\cdot d\vec{s}= B A cos(\theta)=B A cos(\omega t)$
From Faraday's law, induced voltage $V= -\frac{d\Psi}{dt}= BA\omega \sin(\omega t)$
So the voltage is a sine wave (without commutator).
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i agree with the above analysis that AC generator (without commutator) the emf induced in the coil is a sine wave.

so is it that with a commutator, which flip the current in half cycle rotation,  therefore emf induced in the coil will be a |sine| wave?

if that is true, then i understand why the emf induced is |sine| wave.

By the way,
i am certain http://hyperphysics.phy-astr.gsu.edu/HBASE/magnetic/motdct.html is accurate for torque vs time is a |sine| wave.
[img]http://hyperphysics.phy-astr.gsu.edu/HBASE/magnetic/imgmag/dcmtorv.gif[/img]