166 Basic Engineering Mathematics
41 degrees and 29 minutes is written as 41
◦ 29
. 41
◦ 29
is equivalent to 41
29 ◦
60
= 41.483 ◦ as a decimal, correct
to 3 decimal places by calculator.
1 minute further subdivides into 60 seconds,
i.e.
1 minute = 60 seconds
which is written as
1
= 60
.
(Notice that for minutes, 1 dash is used and for seconds,
2 dashes are used.)
For example, 56 degrees, 36 minutes and 13 seconds is
written as 56 ◦ 36 13 .
20.2.2 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. (For
more on circles, see Chapter 26.)
With reference to Figure 20.2, for arc length s,
θ radians =
s
r
r
r
S
O
Figure 20.2
When s is the whole circumference, i.e. when
s = 2πr,
θ =
s
r
=
2πr
r
= 2π
In one revolution, θ = 360 ◦ . Hence, the relationship
between degrees and radians is
360
◦
= 2π radians or 180
◦
= π rad
i.e.
1 rad =
180
◦
π
≈ 57.30
◦
Here are some worked examples on angular measurement.
Problem 1. Evaluate 43 ◦ 29 + 27 ◦ 43
43 ◦ 29
+ 27 ◦ 43
71 ◦ 12
1 ◦
(i) 29 + 43 = 72
(ii) Since 60 = 1 ◦ , 72 = 1 ◦ 12
(iii) The 12 is placed in the minutes column and 1 ◦ is
carried in the degrees column.
(iv) 43 ◦ + 27 ◦ + 1 ◦ (carried) = 71 ◦ . Place 71 ◦ in the
degrees column.
This answer can be obtained using the calculator as
follows.
1. Enter 43
2. Press
◦ ’ ’ ’
3. Enter 29
4. Press ◦ ’ ’ ’
5. Press +
6. Enter 27
7. Press ◦ ’ ’ ’
8. Enter 43
9. Press ◦ ’ ’ ’
10. Press =
Answer = 71
◦ 12
Thus, 43
◦ 29 + 27
◦ 43 = 71
◦ 12 .
Problem 2. Evaluate 84 ◦ 13 − 56 ◦ 39
84 ◦ 13
− 56 ◦ 39
27 ◦ 34
(i) 13 − 39 cannot be done.
(ii) 1
◦ or 60
is ‘borrowed’ from the degrees column,
which leaves 83 ◦ in that column.
(iii) (60 + 13 ) − 39 = 34 , which is placed in the
minutes column.
(iv) 83 ◦ − 56 ◦ = 27 ◦ , which is placed in the degrees
column.
This answer can be obtained using the calculator as
follows.
1. Enter 84
2. Press
◦ ’ ’ ’
3. Enter 13
4. Press ◦ ’ ’ ’
5. Press −
6. Enter 56
7. Press ◦ ’ ’ ’
8. Enter 39
9. Press ◦ ’ ’ ’
10. Press =
Answer = 27
◦ 34
Thus, 84
◦ 13 − 56
◦ 39 = 27
◦ 34 .
Problem 3. Evaluate 19
◦ 51
47
+ 63
◦ 27
34
19 ◦ 51 47
+ 63 ◦ 27 34
83 ◦ 19 21
1 ◦ 1
41 degrees and 29 minutes is written as 41
◦ 29
. 41
◦ 29
is equivalent to 41
29 ◦
60
= 41.483 ◦ as a decimal, correct
to 3 decimal places by calculator.
1 minute further subdivides into 60 seconds,
i.e.
1 minute = 60 seconds
which is written as
1
= 60
.
(Notice that for minutes, 1 dash is used and for seconds,
2 dashes are used.)
For example, 56 degrees, 36 minutes and 13 seconds is
written as 56 ◦ 36 13 .
20.2.2 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. (For
more on circles, see Chapter 26.)
With reference to Figure 20.2, for arc length s,
θ radians =
s
r
r
r
S
O
Figure 20.2
When s is the whole circumference, i.e. when
s = 2πr,
θ =
s
r
=
2πr
r
= 2π
In one revolution, θ = 360 ◦ . Hence, the relationship
between degrees and radians is
360
◦
= 2π radians or 180
◦
= π rad
i.e.
1 rad =
180
◦
π
≈ 57.30
◦
Here are some worked examples on angular measurement.
Problem 1. Evaluate 43 ◦ 29 + 27 ◦ 43
43 ◦ 29
+ 27 ◦ 43
71 ◦ 12
1 ◦
(i) 29 + 43 = 72
(ii) Since 60 = 1 ◦ , 72 = 1 ◦ 12
(iii) The 12 is placed in the minutes column and 1 ◦ is
carried in the degrees column.
(iv) 43 ◦ + 27 ◦ + 1 ◦ (carried) = 71 ◦ . Place 71 ◦ in the
degrees column.
This answer can be obtained using the calculator as
follows.
1. Enter 43
2. Press
◦ ’ ’ ’
3. Enter 29
4. Press ◦ ’ ’ ’
5. Press +
6. Enter 27
7. Press ◦ ’ ’ ’
8. Enter 43
9. Press ◦ ’ ’ ’
10. Press =
Answer = 71
◦ 12
Thus, 43
◦ 29 + 27
◦ 43 = 71
◦ 12 .
Problem 2. Evaluate 84 ◦ 13 − 56 ◦ 39
84 ◦ 13
− 56 ◦ 39
27 ◦ 34
(i) 13 − 39 cannot be done.
(ii) 1
◦ or 60
is ‘borrowed’ from the degrees column,
which leaves 83 ◦ in that column.
(iii) (60 + 13 ) − 39 = 34 , which is placed in the
minutes column.
(iv) 83 ◦ − 56 ◦ = 27 ◦ , which is placed in the degrees
column.
This answer can be obtained using the calculator as
follows.
1. Enter 84
2. Press
◦ ’ ’ ’
3. Enter 13
4. Press ◦ ’ ’ ’
5. Press −
6. Enter 56
7. Press ◦ ’ ’ ’
8. Enter 39
9. Press ◦ ’ ’ ’
10. Press =
Answer = 27
◦ 34
Thus, 84
◦ 13 − 56
◦ 39 = 27
◦ 34 .
Problem 3. Evaluate 19
◦ 51
47
+ 63
◦ 27
34
19 ◦ 51 47
+ 63 ◦ 27 34
83 ◦ 19 21
1 ◦ 1