Trig on the Unit Circle

On a unit circle, the point at angle $\theta$ has coordinates $(\cos\theta, \sin\theta)$. Therefore $\sin^2\theta + \cos^2\theta = 1$ always. Tangent is then $\tan\theta = \frac{\sin\theta}{\cos\theta}$.

Sweep the angle slider from 0 to 360 degrees. The lab moves a point around the unit circle and shows sin, cos, and tan instantly. Try 30, 45, 60, 90 degrees to memorise standard values.

Students swap sin and cos, especially at 30 and 60 degrees. Remember $\sin 30 = \frac{1}{2}$ and $\cos 30 = \frac{\sqrt{3}}{2}$. Tan is undefined at 90 and 270 degrees because cos is zero there.

Carpenters cutting roof angles, photographers framing tall buildings, and surveyors measuring land all use trig. Phone screens use trigonometry for rotation animations - the lab shows the same circle that drives those calculations.