Radians and Degrees

A full turn is equivalent to a turn through 360° and half a turn is equivalent to a turn through 180°. A full turn is also equivalent to a turn throughradians orand half a turn is equivalent to a turn through(Remember that 360° oris the angle at the centre of a circle and 180° oris the angle on a straight line).

We can writeorWe can change radians into degrees using

We also define one minute as one sixtieth of a degree and one second as one sixtieth of a miute. Minutes and seconds are very small angles used when a high degree of precision is required – in astronomy and surveying. Many scientific calculators have the capacity to convert angles measured in decimal degrees and fractions of degrees directly into degrees, minutes and seconds.

The word radian used to measure anges is derived from the word radius referring to the fixed distance of a point on the circumference of a circle from the centre. One radian,is defined as the angle subtended at the centre of a circle by a length of arc equal to the radius.

The circumference of a circle is equal towhich means that there areradians of length in a revolution again telling us that

Using radian to measure angles expresses the angle in terms of arc length.

Trigonometric functions using degrees become circular functions using radians and the domain and range are geometrically the same, but in advanced mathematics it is more natural to use radians.