The unit circle

What you are looking at

On a circle of radius 1 centred at the origin, the point at angle $\theta$ from the positive $x$-axis has coordinates $(\cos\theta, \sin\theta)$. The visualiser draws both coordinates as coloured lines: across is $\cos\theta$, up is $\sin\theta$. A dashed line carries the height of the point to a graph of $0^\circ \le \theta \le 360^\circ$ drawn at the same scale, so the sine wave is traced out as the point goes round. Choosing Draw $\cos\theta$ plots the $x$-coordinate instead.

At $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$, $90^\circ$ and the matching angles in the other quadrants, the readout gives exact values, such as $\cos 135^\circ = -\dfrac{\sqrt{2}}{2}$; elsewhere it gives a decimal. Snapping pulls the point onto those angles. The ASTC letters show which ratios are positive in each quadrant, the tan option draws $\tan\theta$ as a length on the vertical line $x = 1$, and a check line confirms $\sin^2\theta + \cos^2\theta = 1$ at every angle.

Try this

1. 1Drag the point once round the circle and watch the dashed line carry its height across to trace $y = \sin\theta$ from $0^\circ$ to $360^\circ$.
2. 2Switch the labels to Radians and land on $150^\circ$: the readout shows $\dfrac{5\pi}{6}$, $\sin\theta = \dfrac{1}{2}$ and $\cos\theta = -\dfrac{\sqrt{3}}{2}$.
3. 3Turn on ASTC and visit $30^\circ$, $150^\circ$, $210^\circ$ and $330^\circ$: $\sin\theta$ is $\pm\dfrac{1}{2}$ each time, and its sign follows the quadrant letters.
4. 4Turn on tan and move towards $90^\circ$: the tan length on the line $x = 1$ grows without limit, and at exactly $90^\circ$ the readout says undefined.

Mistakes students make

• Swapping sine and cosine. On the unit circle $\cos\theta$ is the $x$-coordinate (across) and $\sin\theta$ is the $y$-coordinate (up), so $\cos 0^\circ = 1$ and $\sin 0^\circ = 0$.
• Losing the sign outside the first quadrant. The reference angle gives the size of the value and the quadrant gives its sign: $\cos 120^\circ = -\dfrac{1}{2}$, not $\dfrac{1}{2}$.
• Leaving the calculator in the wrong mode. $\sin 30 = \dfrac{1}{2}$ only in degrees; in radians $\sin 30 \approx -0.988$, and calculus questions on sine and cosine expect radians.

Where it is taught

• IB Maths AA SL/HL 3.4 · 3.5 · 3.7
• Edexcel A-Level Pure Year 1 · Trigonometry (angles in all four quadrants, exact values, graphs)
• Edexcel A-Level Pure Year 2 · Radians
• Greek Lykeio year 2 (Β΄ Λυκείου) · Algebra, ch. 3 Trigonometry