124 Higher Engineering Mathematics
When s = whole circumference (= 2πr) then
θ =
s
r
=
2πr
r
= 2π
i.e. 2π radians = 360 ◦ or π radians = 180 ◦
Thus, 1rad =
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 3. Convert to radians: (a) 125 ◦
(b) 69 ◦ 47 .
(a) Since 180 ◦ = π rad then 1 ◦ = π/180 rad, therefore
125
◦
= 125
π
180
c = 2.182 rad
(Note that c means ‘circular measure’ and indicates radian measure.)
(b) 69 ◦ 47 = 69
47
◦
60
= 69.783 ◦
69.783
◦
= 69.783
π
180
c = 1.218 rad
Problem 4. Convert to degrees and minutes:
(a) 0.749 rad (b) 3π/4 rad.
(a) Since π rad = 180 ◦ then 1 rad =180 ◦ /π, therefore
0.749 = 0.749
180
π
◦
= 42.915
◦
0.915
◦
= (0.915 × 60)
= 55
, correct to the nearest minute, hence
0.749 rad = 42
◦ 55
(b) Since 1 rad =
180
π
◦
then
3π
4
rad =
3π
4
180
π
◦
=
3
4
(180)
◦
= 135
◦
Problem 5. Express in radians, in terms of π,
(a) 150 ◦ (b) 270 ◦ (c) 37.5 ◦ .
Since 180
◦
= π rad then 1
◦
= 180/π, hence
(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 exercise
Exercise 56 Further problems on radians
and degrees
1. Convert to radians in terms of π: (a) 30 ◦
(b) 75 ◦ (c) 225 ◦ .
(a)
π
6
(b)
5π
12
(c)
5π
4
2. Convert to radians: (a) 48
◦ (b) 84
◦ 51
(c) 232 ◦ 15 .
[(a) 0.838 (b) 1.481 (c) 4.054]
3. Convert to degrees: (a)
5π
6
rad (b)
4π
9
rad
(c)
7π
12
rad.
[(a) 150 ◦ (b) 80 ◦ (c) 105 ◦ ]
4. Convert to degrees and minutes: (a) 0.0125 rad
(b) 2.69 rad (c) 7.241 rad.
[(a) 0 ◦ 43 (b) 154 ◦ 8 (c) 414 ◦ 53 ]
13.4 Arc length and area of circles
and sectors
Arc length
From the definition of the radian in the previous section
and Fig. 13.7,
arc length, s = rθ where θ is in radians
Area of circle
For any circle, area = π × (radius) 2
i.e.
area = πr
2
Since r =
d
2
, then area = πr 2 or
πd
2
4
When s = whole circumference (= 2πr) then
θ =
s
r
=
2πr
r
= 2π
i.e. 2π radians = 360 ◦ or π radians = 180 ◦
Thus, 1rad =
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 3. Convert to radians: (a) 125 ◦
(b) 69 ◦ 47 .
(a) Since 180 ◦ = π rad then 1 ◦ = π/180 rad, therefore
125
◦
= 125
π
180
c = 2.182 rad
(Note that c means ‘circular measure’ and indicates radian measure.)
(b) 69 ◦ 47 = 69
47
◦
60
= 69.783 ◦
69.783
◦
= 69.783
π
180
c = 1.218 rad
Problem 4. Convert to degrees and minutes:
(a) 0.749 rad (b) 3π/4 rad.
(a) Since π rad = 180 ◦ then 1 rad =180 ◦ /π, therefore
0.749 = 0.749
180
π
◦
= 42.915
◦
0.915
◦
= (0.915 × 60)
= 55
, correct to the nearest minute, hence
0.749 rad = 42
◦ 55
(b) Since 1 rad =
180
π
◦
then
3π
4
rad =
3π
4
180
π
◦
=
3
4
(180)
◦
= 135
◦
Problem 5. Express in radians, in terms of π,
(a) 150 ◦ (b) 270 ◦ (c) 37.5 ◦ .
Since 180
◦
= π rad then 1
◦
= 180/π, hence
(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 exercise
Exercise 56 Further problems on radians
and degrees
1. Convert to radians in terms of π: (a) 30 ◦
(b) 75 ◦ (c) 225 ◦ .
(a)
π
6
(b)
5π
12
(c)
5π
4
2. Convert to radians: (a) 48
◦ (b) 84
◦ 51
(c) 232 ◦ 15 .
[(a) 0.838 (b) 1.481 (c) 4.054]
3. Convert to degrees: (a)
5π
6
rad (b)
4π
9
rad
(c)
7π
12
rad.
[(a) 150 ◦ (b) 80 ◦ (c) 105 ◦ ]
4. Convert to degrees and minutes: (a) 0.0125 rad
(b) 2.69 rad (c) 7.241 rad.
[(a) 0 ◦ 43 (b) 154 ◦ 8 (c) 414 ◦ 53 ]
13.4 Arc length and area of circles
and sectors
Arc length
From the definition of the radian in the previous section
and Fig. 13.7,
arc length, s = rθ where θ is in radians
Area of circle
For any circle, area = π × (radius) 2
i.e.
area = πr
2
Since r =
d
2
, then area = πr 2 or
πd
2
4