6.04 Radian measure

From the time of the ancient Babylonians, it has been the practice to divide circles into $360$360 small arcs. An angle can be subtended by extending two lines (rays) from points on the circumference and connecting them at the centre of the circle. The angle subtended at the centre by any one of those $360$360 small 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, it's possible to define the angle subtended by an arc at the centre of a circle, by the ratio of the arc length divided by the radius. Angles defined this way are called radians and an angle in radians can be calculated as $\theta=\frac{s}{r}$θ=sr​, where $s$s is the arc length and $r$r is the radius of the circle. Hence, the angle subtended by an arc whose length is equal to the radius is $1$1 radian. Radians are an alternate way to describe angles and are the international standard unit for measuring angles. Since angles in radian measure are in essence just fractions of the 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 or the abbreviation rad, like this: $2^c$2c or $2$2$rad$rad. (The c is short for circular-measure.)

Restricting attention to circles of radius one unit, $1$1 radian would be the angle subtended by an arc of length $1$1 unit. 

Recall that the circumference of a circle is found using 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. 

Since:

In practice, angles given in radian measure are commonly expressed as fractions of $\pi$π. 

Converting between degrees and radians

Since $\pi^c=180^\circ$πc=180°, to change from degrees to radians divide by $180^\circ$180° and multiply by $\pi$π and vice versa to convert the other way.

Worked example

Example 1 

Convert $\frac{8\pi}{6}$8π6​ to degrees.

Trigonometric functions with radians

Just as the trigonometric functions were explored for angles measured in degrees in the last lesson, these functions can also be applied to angles measured using radians.

Let's look at the sign and symmetry of the trigonometric functions in terms of radians.

Worked example

Example 2

Express $\cos\frac{13\pi}{20}$cos13π20​ in terms of a first quadrant angle.

The angle $\frac{13\pi}{20}$13π20​ is between $\frac{\pi}{2}$π2​ and $\pi$π, so it is in the second quadrant. The point representing $\frac{13\pi}{20}$13π20​ on the unit circle diagram, where the radius cuts the circle, must have a negative horizontal coordinate. Therefore, $\cos\frac{13\pi}{20}$cos13π20​ must be the same as $-\cos\left(\pi-\frac{13\pi}{20}\right)=-\cos\frac{7\pi}{20}$−cos(π−13π20​)=−cos7π20​.

Example 3

A point $P$P on the unit circle is at at angle of $\frac{4\pi}{3}$4π3​ from the positive $x$x-axis, find the coordinates of point $P$P.

The coordinates will be in the form $\left(\cos\theta,\sin\theta\right)$(cosθ,sinθ), so let's use our calculator to evaluate each.

Hence, $P$P is located at $\left(-\frac{1}{2},-\frac{\sqrt{3}}{2}\right)$(−12​,−√32​). We can see that both coordinates are negative since the angle $\frac{4\pi}{3}$4π3​ places point $P$P in the third quadrant.

Practice questions

Question 1

Question 2

Question 3

Question 4