AP Calculus AB and BC glossary

Radian measure

Also called: Radians

Radian measure sizes an angle by the arc it cuts off divided by the radius, so a full circle is two pi radians. Every calculus formula for trigonometric functions assumes radians. In degrees the derivative of sine would be pi over 180 times cosine instead of cosine.

s = r\theta, \quad \lim_{x \to 0}\frac{\sin x}{x} = 1

One radian is the angle that cuts an arc equal in length to the radius, so arc length is $s = r\theta$ and one full turn is $2\pi$ . Because it is a length divided by a length, the measure carries no units, and that is what lets an angle be fed straight into a limit or a series.

The limit $\lim_{x \to 0}\frac{\sin x}{x} = 1$ is the hinge. It is the step that produces $\frac{d}{dx}\sin x = \cos x$ from the definition of the derivative, and it equals 1 only in radians. Measure $x$ in degrees and the same limit is $\frac{\pi}{180}$ , so every trig derivative would drag that constant along and every trig antiderivative would carry its reciprocal.

The mistake

Leaving the calculator in degree mode. The damage is quiet, because the numbers still look reasonable. A degree mode calculator reports $\sin\pi$ as about $0.0548$ instead of 0, so definite integrals, Riemann sums, and slope values on the calculator sections all come out wrong without any error message.

Appears in: Unit 1: Limits and Continuity, Unit 2: Defining the Derivative