Building blocks

## Things you might have noticed

In the diagram below, $P$ is a point on a circle of radius $1$ and $\theta$ is the angle between $OP$ and the positive $x$-axis.

We have drawn in the diameter through $P.$

• $M$ is the point where the diameter through $P$ meets the line $x=1$
• $N$ is the point where the diameter through $P$ meets the line $y=1$

Using the functions from the triangles problem, work out the coordinates of points $P$, $M$ and $N.$

The triangles in the previous part of the problem may help you to work out some side lengths.

We have added some extra labels to the diagram.

The hypotenuse of $OPQ$ has length $1$, so $OQ$ has length $\cos \theta$ and $QP$ has length $\sin \theta$. This means that the coordinates of $P$ are $(\cos \theta, \sin \theta)$.

To find the $y$-coordinate of $M$, note that $OU$ has length $1$ so $UM$ has length $\tan \theta$. Therefore $M$ has coordinates $(1, \tan \theta)$. The similarity of triangles $OPQ$ and $OMU$ again tells us that $\tan \theta = \dfrac{\sin \theta}{\cos \theta}$.

To find the $x$-coordinate of $N$, note that $y=1$ is parallel to the $x$-axis so angle $ONV$ is $\theta.$ As $OV$ has length $1$, this means the coordinates of $N$ are $(\cot \theta, 1)$.

As $\theta$ varies, how will $M$ and $N$ move? What happens when $\theta$ isn’t acute?

• When will $M$ and $N$ be moving quickly?
• Do $M$ and $N$ always move smoothly?
• When will their $x$- or $y$-coordinates change sign?

You can see how $M$ and $N$ move as $\theta$ varies in the GeoGebra file below.

Could you sketch graphs of $\tan \theta$ against $\theta$ and $\cot \theta$ against $\theta$? Download this file to see the graph of $y=\tan \theta$ traced out.

In the next diagram, we have drawn the tangent to the circle at $P$ and marked the points where the tangent at $P$ meets the axes.

What are the coordinates of $K$ and $L$? How will $K$ and $L$ move along the axes as $\theta$ varies?

The line $LK$ is a tangent to the circle at $P$, so $OP$ is at right angles to $LK$.

Therefore triangle $OPK$ is right-angled with angle $\theta$ at $O$ and adjacent side of length $1$. This means that $OK$ has length $\sec \theta$ and the coordinates of $K$ are $(\sec \theta, 0)$.

Likewise, triangle $OPL$ is right-angled with angle $\theta$ at $L$, so $L$ has coordinates $(0,\cosec \theta)$.

You can see how $K$ and $L$ move as $\theta$ varies in the GeoGebra file below. Could you sketch graphs of $\sec \theta$ and $\cosec \theta$ against $\theta$?

We have used $\theta$ to parametrise the circle because as $\theta$ varies, the locus of the point $P(\cos \theta, \sin \theta)$ is a circle, centre $0$, radius $1.$ We could say that $\theta$ parametrises the lines $x=1$ and $y=1$ because the points $M(1, \tan \theta)$ and $N(\cot \theta,1)$ trace out these lines as $\theta$ varies. Similarly, we could say that the $x$ and $y$ axes are parametrised by $\theta$ via the points $K(\sec \theta,0)$ and $L(0, \cosec \theta)$ respectively.