IB Maths AI 3.7 Notes

The unit circle

The unit circle is a circle of radius $1$ centred at the origin. Its equation is:

$x^2+y^2=1$

It is one of the most important ideas in trigonometry because it lets us define $\cos\theta$, $\sin\theta$ and $\tan\theta$ for any angle, not just acute angles in right-angled triangles.

Image description: unit circle with key angles labelled in radians and coordinates shown.

For a point on the unit circle at angle $\theta$, measured anticlockwise from the positive $x$-axis, the coordinates are $(\cos\theta,\sin\theta)$.

So:

• $\cos\theta$ is the $x$-coordinate
• $\sin\theta$ is the $y$-coordinate

Tangent is defined by $\tan\theta=\frac{\sin\theta}{\cos\theta}$. This means $\tan\theta$ is undefined whenever $\cos\theta=0$.