The unit circle: winding a number onto it

The unit circle is the circle with its centre at the origin and radius 1. Its equation:

The circumference of a circle of radius $\displaystyle R$ is $\displaystyle 2\pi R$, so the circumference of the unit circle is $\displaystyle 2\pi$ – about $\displaystyle 6.28$.

On the board: the unit circle, and the points where it meets the axes – $\displaystyle A$, $\displaystyle B$, $\displaystyle C$, $\displaystyle D$.

Every real number $\displaystyle t$ has one point on the unit circle, $\displaystyle P(t)$. To find it, start at the point $\displaystyle A(1,0)$ and walk along the circle an arc of length $\displaystyle |t|$:

• when $\displaystyle t>0$ – anticlockwise;
• when $\displaystyle t<0$ – clockwise;
• when $\displaystyle {t}={0}$ – stay at $\displaystyle A$.

A quarter of the circle is an arc of length $\displaystyle {\frac{2\pi}{4}}={\frac{\pi}{2}}$. So:

• $\displaystyle {P\left(\frac{\pi}{2}\right)}={B(0,1)}$ – a quarter turn;
• $\displaystyle {P(\pi)}={C(-1,0)}$ – half a turn;
• $\displaystyle {P\left(\frac{3\pi}{2}\right)}={D(0,-1)}$ – three quarters;
• $\displaystyle {P(2\pi)}={A(1,0)}$ – a full turn, back to the start;
• $\displaystyle {P\left(-\frac{\pi}{2}\right)}={D(0,-1)}$ – a quarter turn clockwise.

Every point on the circle belongs to infinitely many numbers: one more full turn, either way, brings you back to the same point. If the point belongs to the number $\displaystyle t$, it belongs to all the numbers

where $\displaystyle k$ is an integer ($\displaystyle {k}={0,\ \pm1,\ \pm2,\ \dots}$).

• $\displaystyle {k}={1}$ – one more turn anticlockwise;
• $\displaystyle {k}={-1}$ – one turn clockwise;
• $\displaystyle {k}={3}$ – three more turns.

The example below finds the points $\displaystyle P\left(\frac{9\pi}{4}\right)$ and $\displaystyle P\left(-\frac{5\pi}{2}\right)$.

• $\displaystyle {\frac{9\pi}{4}}={\frac{\pi}{4}+2\pi}$: a full turn and then $\displaystyle \frac{\pi}{4}$ more – halfway from $\displaystyle A$ to $\displaystyle B$. This is the point $\displaystyle P$ on the board, in the middle of the arc in the first quadrant.
• $\displaystyle {-\frac{5\pi}{2}}={-\frac{\pi}{2}-2\pi}$: a full turn clockwise and a quarter turn more. This is the point $\displaystyle {Q}={D(0,-1)}$.

The axes split the circle into four quadrants. A number between $\displaystyle 0$ and $\displaystyle 2\pi$ is:

• between $\displaystyle 0$ and $\displaystyle \frac{\pi}{2}$ – in the first quadrant;
• between $\displaystyle \frac{\pi}{2}$ and $\displaystyle \pi$ – in the second quadrant;
• between $\displaystyle \pi$ and $\displaystyle \frac{3\pi}{2}$ – in the third quadrant;
• between $\displaystyle \frac{3\pi}{2}$ and $\displaystyle 2\pi$ – in the fourth quadrant.

A number not written with $\displaystyle \pi$ is compared with $\displaystyle \frac{\pi}{2}\approx1.57$, $\displaystyle \pi\approx3.14$, $\displaystyle \frac{3\pi}{2}\approx4.71$, $\displaystyle 2\pi\approx6.28$. For example, $\displaystyle {t}={2}$ is between $\displaystyle 1.57$ and $\displaystyle 3.14$, so $\displaystyle P(2)$ is in the second quadrant.

All the numbers that belong to the same point as $\displaystyle t$ are $\displaystyle {x}={t+2\pi k}$.
Find the one of them that is between $\displaystyle 0$ and $\displaystyle 2\pi$.

Which quadrant is the point in?
$\displaystyle P(4)$, $\displaystyle P(-1)$, $\displaystyle P(10)$.
Write the quadrant of each point, and how you found it.

An ant walks anticlockwise on the unit circle, starting at the point $\displaystyle A(1,0)$.
It walks exactly $\displaystyle 20$ units of length.

How many turns does it make?
Round to two decimal places (the last line starts with $\displaystyle \approx$).

In which quadrant does it stop?