AP Calculus AB and BC

Radians vs Degrees in Calculus

Q: What is the derivative of sin x in degrees?

\frac{\pi}{180}\cos x. The extra factor comes from the chain rule converting degrees to radians.

Calculus uses radians, always. The derivative of sin x is cos x only in radians; in degrees it is pi over 180 times cos x. The reason is the special limit sin x over x, which equals 1 in radians and pi over 180 in degrees, and every trig derivative is built on it.

Radians

Use when: Always, in every calculus context: derivatives, integrals, limits, series, and any calculator work on the exam.

Degrees

Use when: Never in calculus. Only in geometry or when a problem explicitly reports an angle in degrees for interpretation.

Side by side

	Radians	Degrees
$\lim_{x \to 0}\frac{\sin x}{x}$	$1$	$\frac{\pi}{180} \approx 0.01745$
$\frac{d}{dx}\sin x$	$\cos x$	$\frac{\pi}{180}\cos x$
Formula sheet assumes	Radians	Never
Calculator mode for the exam	Radian	Wrong answers
$\sin\left(\frac{\pi}{6}\right)$	$0.5$	About $0.00914$

The last row is the practical failure. A calculator left in degree mode evaluates $\sin\left(\frac{\pi}{6}\right)$ as the sine of about $0.524$ DEGREES, which is roughly $0.00914$ instead of $0.5$ . Every downstream number is then wrong, and nothing on the page looks obviously broken.

Why radians are the natural unit

A radian is defined so the arc length on a unit circle equals the angle. That is exactly what makes sin x behave like x near zero, and it is why every derivative formula comes out clean. Degrees are an arbitrary division of a circle into 360 parts, and the factor of pi over 180 is the price.

Frequently asked questions

Does calculus use radians or degrees?

Radians, always. Every derivative and limit formula assumes them.

What is the derivative of sin x in degrees?

$\frac{\pi}{180}\cos x$ . The extra factor comes from the chain rule converting degrees to radians.

What happens if my calculator is in degree mode?

Every trigonometric value is wrong, usually by a large factor, and the error propagates silently. Check the mode before the exam starts.

In the CED: Unit 1: Limits and Continuity, Unit 2: Defining the Derivative