1 Trigonometric functions for any size angle

1.1 The radian

First we introduce an alternative to measuring angles in degrees. Look at the circle shown in Figure 19(a). It has radius $r$ and we have shown an arc $AB$ of length $\ell$ (measured in the same units as $r$ .) As you can see the arc subtends an angle $\theta$ at the centre O of the circle.

Figure 19

The angle $\theta$ in radians is defined as

$\theta =\frac{\text{length of arc}AB}{\text{radius}}=\frac{\ell }{r}$

So, for example, if $r=10$ cm, $\ell =20$ cm, the angle $\theta$ would be $\frac{20}{10}=2$ radians.

The relation between the value of an angle in radians and its value in degrees is readily obtained as follows. Referring to Figure 19(b) imagine that the arc $AB$ extends to cover half the complete perimeter of the circle. The arc length is now $\pi r$ (half the circumference of the circle) so the angle $\theta$ subtended by $AB$ is now

$\theta =\frac{\pi r}{r}=\pi \text{radians}$

But clearly this angle is $180^{\circ }$ . Thus $\pi$ radians is the same as $180^{\circ }$ .

Note conversely that since $\pi$ radians $=180^{\circ }$ then 1 radian $=\frac{180}{\pi }$ degrees (about $57.3^{\circ }$ ).

Task!

Write down the values in radians of $30^{\circ },45^{\circ },90^{\circ },135^{\circ }$ . (Leave your answers as multiples of $\pi$ .)

Answer

Task!

Write in degrees the following angles given in radians

$\frac{\pi }{10},\frac{\pi }{5},\frac{7\pi }{10},\frac{23\pi }{12}$

Answer

Task!

Put your calculator into radian mode (using the DRG button if necessary) for this Task: Verify these facts by first converting the angles to radians:

$sin30^{\circ }=\frac{1}{2}cos45^{\circ }=\frac{1}{\sqrt{2}}tan60^{\circ }=\sqrt{3}$ (Use the $\pi$ button to obtain $\pi$ .)

Answer