5.1PureCore

Radian Measure

A radian is a more natural unit for measuring angles than the degree. One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. Radians make arc length, sector area, and calculus of trig functions far simpler.

25 min Video by Zeeshan Zamurred Radians

Edexcel A Level Maths: 5.1 Radians (Angles)Watch the full walkthrough before the notes below.

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What you'll be able to do

Define a radian
Convert between degrees and radians
Know common angles in radians
Evaluate trig functions of radian angles

Degrees ↔ radians

A full turn is $360°$ or $2 π$ radians, so $180° = π$ radians. To convert degrees to radians multiply by $\frac{π}{180}$ ; to convert radians to degrees multiply by $\frac{180}{π}$ .

180° = π rad

The key conversion fact.

Common angles

Memorise: $30° = \frac{π}{6}$ , $45° = \frac{π}{4}$ , $60° = \frac{π}{3}$ , $90° = \frac{π}{2}$ , $360° = 2 π$ .

135 \times \frac{π}{180} = \frac{135 π}{180}

2Simplify:

\frac{3 π}{4}

Answer

\frac{3 π}{4}

radians

Tip — Keep answers as exact multiples of π unless told to use decimals.

Trig of radian angles

Trig functions take radian inputs directly: e.g. $sin \frac{π}{6} = \frac{1}{2}$ . In exams, set your calculator to radian mode when angles are in radians.

Formula recap

180° = π rad

Base conversion.

deg \to rad : \times \frac{π}{180}

Degrees to radians.

rad \to deg : \times \frac{180}{π}

Radians to degrees.

Common mistakes to avoid

Leaving the calculator in degree mode for radian angles.

Switch to radian (RAD) mode when the angle is in radians.

Converting with the wrong factor.

Degrees→radians multiply by π/180; radians→degrees multiply by 180/π.

Key takeaways

180° = π radians is the key fact.
deg→rad ×π/180; rad→deg ×180/π.
Know π/6, π/4, π/3, π/2 by heart.

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