AHL 3.8—Unit circle, Pythag identity, solving trig equations graphically

The unit circle is one of the most powerful tools in trigonometry. It connects angles to coordinates in a beautifully simple way.

Unit Circle: A circle with radius 1 centred at the origin $(0,0)$ in the Cartesian plane. Its equation is $x^2+y^2=1$.

For any angle $\theta$ measured from the positive $x$-axis (anticlockwise), the point where the terminal side of the angle meets the unit circle has coordinates:

$P=(\cos \theta ,\sin \theta )$

This means:

• The x-coordinate of the point equals $\cos \theta$
• The y-coordinate of the point equals $\sin \theta$
• The slope of the radius to that point equals $\tan \theta =\frac{\sin \theta }{\cos \theta }$

Because the radius is always 1, we don't need to divide by a hypotenuse , the coordinates are the trigonometric ratios.

Because $\sin \theta$ and $\cos \theta$ are coordinates on the unit circle, their signs depend entirely on which quadrant the angle $\theta$ falls in.

Notice that $\tan \theta =\sin \theta /\cos \theta$, so it is positive whenever both have the same sign (Q1 and Q3) and negative when they differ (Q2 and Q4).