P1^
7
Circular(^) measure
ExAmPlE 7.9 (i) Express in radians (a) 30° (b) 315° (c) 29°.
(ii) Express in degrees (a) π
12
(b) 8
3
π (c) 1.2 radians.
SOlUTION
(i) (a) 30° = 30 × ππ
180 6
=
(b) 315° = 315 × ππ
1807
4
=
(c) 29° = 29 × π(^180)
= 0.506 radians (to 3 s.f.).
(ii) (a) ππ
12 12 π
=×^180
= 15°
(b) 83 ππ=×^83180 π
= 480°(c) 1.2 radians = 1.2 × (^180) π = 68.8° (to 3 s.f.).
Using your calculator in radian mode
If you wish to find the value of, say, sin 1.4c or cos 12 π, use the ‘rad’ mode on your
calculator. This will give the answers directly − in these examples 0.9854... and
0.9659....
You could alternatively convert the angles into degrees (by multiplying by^180 π)
but this would usually be a clumsy method. It is much better to get into the habit
of working in radians.
ExAmPlE 7.10 Solve sin θ = 12 for 0 θ 2 π giving your answers as multiples of π.
SOlUTION
Since the answers are required as multiples of π it is easier to work in degrees first.
sin θ = ^12
⇒ θ = 30°
θ = 30 ×
ππ
180 6
=.
From figure 7.25 there is a second value
θ = 150° =5
6
π
.The values of θ are ππ 6 and^56.O 180° 360° θ(^12)
sin θ
Figure 7.25