Why use radians instead of degrees?

Radians are defined by the geometry of the circle, so calculus results and arc-length formulas come out without extra conversion factors.

How do I convert degrees to radians?

Multiply by π and divide by 180 — for example 45° becomes π/4.

MathematicsA Level & IB

A radian is the angle subtended at the centre of a circle by an arc equal in length to the radius; 2π radians make a full turn.

Radians make arc length and sector area formulas simple, and they are required for calculus with trigonometric functions — the derivative of

\sin x

is only

\cos x

in radians.

Convert with

180^{\circ} = \pi

radians, and leave answers in terms of

\pi

when the question asks for exact values.

s = r\theta, \qquad A = \tfrac{1}{2}r^{2}\theta

A sector with

r = 6

and

\theta = \frac{\pi}{3}

has arc length

2\pi

and area

6\pi

Switch your calculator to radian mode at the start of any A Level trigonometry or calculus question and check the display indicator.

Why use radians instead of degrees?: Radians are defined by the geometry of the circle, so calculus results and arc-length formulas come out without extra conversion factors.
How do I convert degrees to radians?: Multiply by π and divide by 180 — for example 45° becomes π/4.

Trigonometry relates the angles of a triangle to the lengths of its sides using the sine, cosine and tangent ratios.

Differentiation is the process of finding the rate at which one quantity changes with respect to another — the gradient of a curve at a point.

Take radian from a definition to full marks with exam-board question banks, revision notes and instant step-by-step feedback from Study Coach.

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