From the time of the ancient Babylonians, it has been the practice to divide circles into $360$360 small arcs. The angle subtended at the centre by any one of those arcs, is called one degree. In effect, an arc of the circle is used as a measure of the angle it subtends.
In a similar way, we now restrict our attention to circles of radius one unit, this is called the unit circle. We measure angles subtended at the centre by arcs of this circle. This method of measuring angles is called radian measure.
Do you remember how to find the circumference of a circle? We use the formula $C=2\pi r$C=2πr. So if the radius ($r$r) is $1$1, then the circumference is $2\pi$2π.
The angle represented by a full turn around the circle is $2\pi$2π radians. This is equivalent to $360^\circ$360°.
A half-circle makes an angle of $\pi$π radians or $180^\circ$180°
and a right-angle is $\frac{\pi}{2}$π2 radians.
An angle of $1$1 radian must be $\frac{360^\circ}{2\pi}\approx57.3^\circ$360°2π≈57.3° .
In practice, angles given in radian measure are usually expressed as fractions of $\pi$π.
Because angles in radian measure are in essence just fractions of the unit circle, they do not require a unit, although some writers indicate that radian measure is being used by adding a superscript c after a number, like this: $\frac{\pi}{6}^{^c}$π6c . (The c is short for circular-measure.)
Trigonometric functions of a variable angle measured in radians can thus be thought of, like other functions, as functions of a real number. Thus, we encounter function definitions like $y=\sin x$y=sinx, where $x$x is understood to be an ordinary number.
Worked Examples
QUESTION 1
Convert $90^\circ$90° to radians.
Give your answer in exact form.
QUESTION 2
Convert $-300^\circ$−300° to radians.
Give your answer in exact form.
QUESTION 3
Convert $\frac{2\pi}{3}$2π3 radians to degrees.
QUESTION 4
Convert $4.2$4.2 radians to degrees.
Give your answer correct to one decimal place.