Deriving equation for cycloid

Question

Here, do we measure angle $t$ from $\vec{TC}$ to $\vec{CP}$ ; or from $\vec{CP}$ to $\vec{TC}$? In particular, I am finding it very hard to digest that coordinates of vector $\vec{CP}$ are $(-a \sin t,-a \cos t)$, which is discussed a couple minutes later in linked video. Can you elaborate, or provide another viewpoint?

Answer 1

Angle $t$ is measured clockwise, from $T$ to $P$. That's consistent with the right-wise motion of the rolling circle: $t$ increases during motion.

Vector $\vec{CP}$ has negative coordinates for $0<t<\pi/2$, so it is not surprising if a minus sign is needed.