DEGREE MEASURE OF AN ANGLE

The degree is a unit of measurement of angles and is represented by the symbol °.

In degrees, we split up one complete rotation into 360 equal parts and each part is one degree, denoted by 1°. Thus, 1° is 1/360 of one complete rotation.

To measure a fraction of an angle and also for accuracy of measurement of angles, minutes and seconds are introduced. One minute (1') corresponds to 1/60 of a degree and in turn a second (1'') corresponds to 1/60 of a minute (or) 1/3600 of a degree.

Classification of Pair of Angles

We shall classify a pair of angles in the following way for better understanding and usages.

(i) Two angles that have the exact same measure are called congruent angles.

(ii) Two angles that have their measures adding to 90° are called complementary angles.

(iii) Two angles that have their measures adding to 180° are called supplementary angles.

(iv) Two angles between 0° and 360° are conjugate if their sum equals 360°.

Degrees, Minutes and Seconds

(i) The concept of degrees, minutes and seconds, is analogous to the system of time measurement where we think of a degree representing one hour.

(ii) Observe that

59.0854° = 59° + 0.0854°

0.0854° = 0.0854° ⋅ (60'/1°) = 5.124'

5.124' = 5' + 0.124'

0.124' = 0.124' ⋅ (60"/1') = 7.44"

Thus, 

59.0854° = 59°5'7.44"

(iii) Also notice that

34°51'35" = 34.8597°

and

90° - 36°18'47" = 53°41'13"

Convert each decimal angle measure into degrees, minutes & seconds.

Problem 1 :

25.64°

Solution :

25.64° = 25 + 0.64

1° = 60 minutes

= 25 + 0.64(60)

= 25 + 38.4

1' = 60 seconds

= 25 degree + 38 minutes + 0.4 (60)

= 25 degree + 38 minutes + 24 seconds

 25° 38' 24''

Problem 2 :

136.28°

Solution :

136.28° = 136 + 0.28

= 136 + 0.28(60)

= 136 + 16.8

= 136 degree + 16 minutes + 0.8 (60)

= 136 degree + 16 minutes + 48 seconds

= 136° 16' 48''

Problem 3 :

359.96°

Solution :

359.96° = 359 + 0.96

= 359 + 0.96(60)

= 359 + 57.6

= 359 degree + 57 minutes + 0.6 (60)

= 359 degree + 57 minutes + 36 seconds

= 359° 57' 36''

Problem 4 :

204.56°

Solution :

204.56° = 204 + 0.56

= 204 + 0.56(60)

= 204 + 33.6

= 204 degree + 33 minutes + 0.6 (60)

= 204 degree + 33 minutes + 36 seconds

= 204° 33' 36''

Problem 5 :

300.26°

Solution :

300.26° = 300 + 0.26

= 300 + 0.26(60)

= 300 + 15.6

= 300 degree + 15 minutes + 0.6 (60)

= 300 degree + 15 minutes + 36 seconds

= 300° 15' 36''

Convert each angle measure in degrees, minutes & seconds to decimal degrees.

Problem 6 :

49 degrees, 46 minutes, 48 seconds

Solution :

49 degrees, 46 minutes, 48 seconds

1° = 60 minutes

1 minute = 1/60 degree

1 minute = 60 seconds 

1 second = 1/3600 degree

46 minutes = 46/60

= 0.766

48 seconds = 48/3600

= 0.0133

0.766 + 0.0133 = 0.7793

Approximately 0.78

49 degrees, 46 minutes, 48 seconds = 49.78°

Problem 7 :

98 degrees, 45 minutes, 0 seconds

Solution :

98 degrees, 45 minutes, 0 seconds

1° = 60 minutes

45 minutes = 45/60

= 0.75

0 seconds = 0/3600

= 0

0.766 + 0 = 0.75

Approximately 0.75

98 degrees, 45 minutes, 0 seconds = 98.75°

Problem 8 :

135 degrees, 41 minutes, 24 seconds

Solution :

135 degrees, 41 minutes, 24 seconds

1° = 60 minutes

41 minutes = 41/60

= 0.683

24 seconds = 24/3600

= 0.006

0.683 + 0.006 = 0.689

Approximately 0.69

135 degrees, 41 minutes, 24 seconds = 135.69°

Problem 9 :

167 degrees, 51 minutes, 36 seconds

Solution :

167 degrees, 51 minutes, 36 seconds

1° = 60 minutes

51 minutes = 51/60

= 0.85

36 seconds = 36/3600

= 0.01

0.85 + 0.01 = 0.86

Approximately 0.86

167 degrees, 51 minutes, 36 seconds = 167.86°

Problem 10 :

210 degrees, 34 minutes, 48 seconds

Solution :

210 degrees, 34 minutes, 48 seconds

1° = 60 minutes

34 minutes = 34/60

= 0.566

48 seconds = 48/3600

= 0.0133

0.566 + 0.013 = 0.579

Approximately 0.58

210 degrees, 34 minutes, 48 seconds = 210.58°

Problem 11 :

2 degrees, 49 minutes, 12 seconds

Solution :

2 degrees, 49 minutes, 12 seconds

1° = 60 minutes

49 minutes = 49/60

= 0.816

12 seconds = 12/3600

= 0.003

0.816 + 0.003 = 0.819

Approximately 0.82

2 degrees, 49 minutes, 12 seconds = 2.82°

Problem 12 :

Convert each angle from degrees to radians, giving your answers to 2 decimal places.

a) 10°

b) 38°

c) 291°

Solution :

a)

10° = 10 x (π /180)

= π/18

= 3.14/18

= 0.174

Approximately 0.17.

b)

38° = 38 x (π /180)

= 19π/90

= 19(3.14)/90

= 0.662

Approximately 0.66.

c)

291° = 291 x (π /180)

= 291π/180

= 291(3.14)/180

= 5.076

Approximately 5.08

Related Pages

Angles in standard position

Coterminal angles

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