232 Basic Engineering Mathematics
7. Calculate the length of wire in the paper clip
shown in Figure 26.6. The dimensions are in
millimetres.
2.5 rad
2.5 rad
3 rad
12
6
32
Figure 26.6
26.3 Radians and degrees
One radian is defined as the angle subtended at the
centre of a circle by an arc equal in length to the radius.
With reference to Figure 26.7, for arc length s,
θ radians =
s
r
s
r
O
r
Figure 26.7
When s = whole circumference (= 2πr) then
θ =
s
r
=
2πr
r
= 2π
i.e. 2π radians = 360
◦ or π radians = 180
◦
Thus, 1 rad =
180 ◦
π
= 57.30
◦
, correct to 2 decimal
places.
Since π rad = 180 ◦ , then
π
2
= 90 ◦ ,
π
3
= 60 ◦ ,
π
4
= 45 ◦ ,
and so on.
Problem 5. Convert to radians: (a) 125 ◦
(b) 69
◦ 47
(a) Since 180 ◦ = π rad, 1 ◦ =
180
π
rad, therefore
125
◦
= 125
π
180
rad = 2.182 radians.
(b) 69 ◦ 47 = 69
47 ◦
60
= 69.783 ◦ (or, with your calculator, enter 69 ◦ 47 using ◦ ’ ’ ’ function, press =
and press ◦ ’ ’ ’ again).
and 69.783 ◦ = 69.783
π
180
rad
= 1.218 radians.
Problem 6. Convert to degrees and minutes:
(a) 0.749 radians (b) 3π/4 radians
(a) Since π rad = 180
◦
, 1 rad =
180 ◦
π
therefore 0.749 rad= 0.749
180
π
◦
= 42.915 ◦
0.915 ◦ = (0.915 × 60) = 55 , correct to the nearest minute,
Hence, 0.749 radians = 42 ◦ 55
(b) Since 1 rad =
180
π
o
then
3π
4
rad =
3π
4
180
π
o
=
3
4
(180) ◦ = 135 ◦
Problem 7. Express in radians, in terms of π,
(a) 150 ◦ (b) 270 ◦ (c) 37.5 ◦
Since 180 ◦ = πrad, 1 ◦ =
π
180
rad
(a) 150 ◦ = 150
π
180
rad =
5π
6
rad
(b) 270 ◦ = 270
π
180
rad =
3π
2
rad
(c) 37.5 ◦ = 37.5
π
180
rad =
75π
360
rad =
5π
24
rad
Now try the following Practice Exercise
Practice Exercise 102 Radians and degrees
(answers on page 351)
1. Convert to radians in terms of π:
(a) 30
◦
(b) 75
◦
(c) 225
◦
2. Convert to radians, correct to 3 decimal
places:
(a) 48 ◦ (b) 84 ◦ 51 (c) 232 ◦ 15
7. Calculate the length of wire in the paper clip
shown in Figure 26.6. The dimensions are in
millimetres.
2.5 rad
2.5 rad
3 rad
12
6
32
Figure 26.6
26.3 Radians and degrees
One radian is defined as the angle subtended at the
centre of a circle by an arc equal in length to the radius.
With reference to Figure 26.7, for arc length s,
θ radians =
s
r
s
r
O
r
Figure 26.7
When s = whole circumference (= 2πr) then
θ =
s
r
=
2πr
r
= 2π
i.e. 2π radians = 360
◦ or π radians = 180
◦
Thus, 1 rad =
180 ◦
π
= 57.30
◦
, correct to 2 decimal
places.
Since π rad = 180 ◦ , then
π
2
= 90 ◦ ,
π
3
= 60 ◦ ,
π
4
= 45 ◦ ,
and so on.
Problem 5. Convert to radians: (a) 125 ◦
(b) 69
◦ 47
(a) Since 180 ◦ = π rad, 1 ◦ =
180
π
rad, therefore
125
◦
= 125
π
180
rad = 2.182 radians.
(b) 69 ◦ 47 = 69
47 ◦
60
= 69.783 ◦ (or, with your calculator, enter 69 ◦ 47 using ◦ ’ ’ ’ function, press =
and press ◦ ’ ’ ’ again).
and 69.783 ◦ = 69.783
π
180
rad
= 1.218 radians.
Problem 6. Convert to degrees and minutes:
(a) 0.749 radians (b) 3π/4 radians
(a) Since π rad = 180
◦
, 1 rad =
180 ◦
π
therefore 0.749 rad= 0.749
180
π
◦
= 42.915 ◦
0.915 ◦ = (0.915 × 60) = 55 , correct to the nearest minute,
Hence, 0.749 radians = 42 ◦ 55
(b) Since 1 rad =
180
π
o
then
3π
4
rad =
3π
4
180
π
o
=
3
4
(180) ◦ = 135 ◦
Problem 7. Express in radians, in terms of π,
(a) 150 ◦ (b) 270 ◦ (c) 37.5 ◦
Since 180 ◦ = πrad, 1 ◦ =
π
180
rad
(a) 150 ◦ = 150
π
180
rad =
5π
6
rad
(b) 270 ◦ = 270
π
180
rad =
3π
2
rad
(c) 37.5 ◦ = 37.5
π
180
rad =
75π
360
rad =
5π
24
rad
Now try the following Practice Exercise
Practice Exercise 102 Radians and degrees
(answers on page 351)
1. Convert to radians in terms of π:
(a) 30
◦
(b) 75
◦
(c) 225
◦
2. Convert to radians, correct to 3 decimal
places:
(a) 48 ◦ (b) 84 ◦ 51 (c) 232 ◦ 15