AP Calculus AB and BC glossary

Unit Circle

Also called: Unit circle chart

The unit circle is the circle of radius 1 centred at the origin, where travelling a signed arc length t from the point (1, 0), counterclockwise when t is positive and clockwise when t is negative, lands you at the point (cos t, sin t). It supplies the exact trig values calculus expects, with t measured in radians.

x^2 + y^2 = 1

Every point on the circle is $(\cos t, \sin t)$ , so the coordinates are the trig values themselves rather than something you compute from them. The three first quadrant inputs worth knowing cold are $\frac{\pi}{6}$ , $\frac{\pi}{4}$ and $\frac{\pi}{3}$ , whose sines are $\frac{1}{2}$ , $\frac{\sqrt{2}}{2}$ and $\frac{\sqrt{3}}{2}$ , with the cosines running that same list backwards. Reflecting into the other three quadrants supplies the signs.

(\cos t,\ \sin t) \quad \text{with} \quad \cos^2 t + \sin^2 t = 1

Signed arc length is what makes negative inputs behave. Going clockwise by $t$ lands on the mirror image across the $x$ axis of the counterclockwise point, which is the picture behind $\cos(-t) = \cos t$ and $\sin(-t) = -\sin t$ , and it is why a left hand limit at $0$ reads off the same circle as the right hand one. The other reflections do the same work: $\sin(\pi - t) = \sin t$ and $\cos(\pi + t) = -\cos t$ are two readings of one point, not identities to store separately. Because $t$ is arc length on a radius $1$ circle, it is a radian measure, which is the form every derivative rule for sine and cosine assumes.

The mistake

Reading a point as $(\sin t, \cos t)$ . The first coordinate is the cosine: at $t = \frac{\pi}{6}$ the point is $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$ , so $\sin\frac{\pi}{6} = \frac{1}{2}$ and $\cos\frac{\pi}{6} = \frac{\sqrt{3}}{2}$ . Swapping them reflects the whole circle across the line $y = x$ , which trades those two values and mirrors the sign pattern of every quadrant.

Appears in: Unit 1: Limits and Continuity