# Rotating derivatives Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

## Suggestion

Particle $P$ is moving anti-clockwise round a circle of radius $\quantity{1}{m}$ at speed $\quantity{1}{m\,s^{-1}}$.

The particle is at point $A$ when $t=0$ and its position at time $\quantity{t}{s}$ is shown in the diagram below.

Combine some of the following diagrams to explain why the velocity vector $\mathbf{v}$ is $(-\sin t, \cos t)$. You may be able to do this in more than one way.

• How have you used the fact that $P$ is moving at unit speed on a unit circle?

• What does this tell you about the derivatives of $\sin t$ and $\cos t$?

Here are some questions to ask yourself when looking at the diagrams. (The diagrams with questions are also provided on cards.)

What is the magnitude and direction of $\mathbf{v}$?

Why is $P$ in this position at time $\quantity{t}{s}$?

How are vectors $\mathbf{v}$ and $\overrightarrow{OQ}$ related?

How are the two triangles related?

Why is $\mathbf{v}$ given by the vector shown?

Why are these the coordinates of $Q$?