Unit Circle

The unit circle is the circle of radius 1 centred at the origin.

For any angle $\theta$ measured COUNTER-CLOCKWISE from the positive x-axis, the corresponding point on the circle has coordinates:

$(\cos \theta, \sin \theta).$

So $\cos \theta$ is the x-coordinate, and $\sin \theta$ is the y-coordinate.

This extends $\sin$ and $\cos$ to ALL angles, including those greater than $90°$ or negative.

Worked example. Find $\cos 90°$ and $\sin 90°$.

• At $90°$ (counter-clockwise from positive x-axis), the point on the unit circle is $(0, 1)$.
• So $\cos 90° = 0$ and $\sin 90° = 1$.

Worked example. Find $\cos 180°$.

• Point at $180°$: $(-1, 0)$. So $\cos 180° = -1$.

Worked example. Find $\sin 270°$.

• Point at $270°$: $(0, -1)$. So $\sin 270° = -1$.

Tangent on the unit circle: $\tan \theta = \dfrac{\sin \theta}{\cos \theta}$ — the ratio of y to x. UNDEFINED when $\cos \theta = 0$ (at $90°, 270°$).

As $\theta$ moves around the unit circle, the signs of $\sin, \cos, \tan$ change.

Quadrant	Angle range	$\sin$	$\cos$	$\tan$
Q1	$0°-90°$	$+$	$+$	$+$
Q2	$90°-180°$	$+$	$-$	$-$
Q3	$180°-270°$	$-$	$-$	$+$
Q4	$270°-360°$	$-$	$+$	$-$

The CAST rule (or ASTC, depending on direction read): each quadrant has ONE trig ratio that's positive (apart from Q1 where ALL are positive).

• Q1: All positive.
• Q2: Sin positive.
• Q3: Tan positive.
• Q4: Cos positive.

Mnemonic: "All Students Take Calculus" (Q1, Q2, Q3, Q4).

Worked example. $\theta = 210°$. Find $\sin 210°$.

• $210°$ is in Q3.
• $\sin$ is NEGATIVE in Q3.
• Reference angle: $210° - 180° = 30°$.
• $|\sin 30°| = 1/2$, with negative sign in Q3: $\sin 210° = -1/2$.

Reference angle: the acute angle made with the nearest part of the x-axis. Use it to find the magnitude of the trig ratio, then attach the sign based on the quadrant.