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angles, respectively, and so on. By the continued revolution of OP the angle between the initial line OA and the revolving line OP may become of any magnitude whatever.
In the same way OP may revolve in the negative direction about O any number of times, generating a negative angle; and this negative angle may obviously have any magnitude whatever.
The angle AOP may be the geometric representative of any of the Trigonometric angles formed by any number of complete revolutions, either in the positive direction added to the positive angle AOP, or in the negative direction added to the negative angle AOP. In all cases the angle is said to be in the quadrant indicated by its terminal line.
There are three methods of measuring angles, called respectively the Sexagesimal, the Centesimal, and the Circular methods.
6. The Sexagesimal Method. — This is the method in general use. In this method the right angle is divided into 90 equal parts, each of which is called a degree. Each degree is subdivided into 60 equal parts, each of which is called a minute. Each minute is subdivided into 60 equal parts, each of which is called a second. Then the magnitude of an angle is expressed by the number of degrees, minutes, and seconds which it contains. Degrees, minutes, and seconds are denoted respectively by the symbols , ', ": thus, to represent 18 degrees, 6 minutes, 34.58 seconds, we write
18° 6' 34".58.
A degree of arc is go of the circumference to which the arc belongs. The degree of arc is subdivided in the same manner as the degree of angle. Then 1 circumference 360° 21600' =1296000".
1 quadrant or right angle = 90°. . Instruments used for measuring angles are subdivided accordingly.
7. The Centesimal or Decimal Method. - In this method the right angle is divided into 100 equal parts, each of which is called a grade. Each grade is subdivided into 100 equal parts, each of which is called a minute. Each minute is subdivided into 100 equal parts, each of which is called a second. The magnitude of an angle is then expressed by the number of grades, minutes, and seconds which it contains. Grades, minutes, and seconds are denoted respectively by the symbols 5,, ": thus, to represent 34 grades, 48 minutes, 86.47 seconds, we write
348 48' 86'\.47.
The centesimal or decimal method was proposed by the French mathematicians in the beginning of the present century. But although it possesses many advantages over the established method, they were not considered sufficient to counterbalance the enormous labor which would have been necessary to rearrange all the mathematical tables, books of reference, and records of observations, which would have to be transferred into the decimal system before its advantages could be felt. Thus, the centesimal method has never been used even in France, and in all probability never will be used in practical work.
8. The Circular Measure. — The unit of
B circular measure is the angle subtended at the centre of a circle by an arc equal in length to the radius.
This unit of circular measure is called a radian.
Let O be the centre of a circle whose radius is r.
Let the arc AB be equal to the radius OA =
B Then, since angles at the centre of a circle are in the same ratio as their intercepted arcs (Geom., Art. 234), and since the ratio of the circumference of a circle to its diameter is = 3.14159265 (Geom., Art. 436),
.: angle AOB : 4 rt. angles :: arc AB : circumference,
4 rt. angles 2 rt. angles .: angle AOB
180° .. a radian = angle AOB=
57°.2957795 3.14159265 = 3437'.74677 = 206264".806.
Therefore, the radian is the same for all circles, and = 57°.2957795.
A Then arc AB = radius OA. .. angle AOP : angle AOB
:: arc AP: arc AB; or angle AOP : radian :: arc AP : radius. .: angle AOP=
x radian. radius
The measure of any quantity is the number of times it contains the unit of measure (Art. 2). .:: the circular measure of angle AOP
NOTE 1. – The student will notice that a radian is a little less than an angle of an equilateral triangle, i.e., of 60°.
Angles expressed in circular measure are usually denoted by Greek letters, a, b, Y, ..., 0, 0, 4,
The circular measure is employed in the various branches of Analytical Mathematics, in which the angle under consideration is almost always expressed by a letter.
NOTE 2. – The student cannot too carefully notice that unless an angle is obvi. ously referred
the letters a, b, 0, 0, ... stand for mere numbers. Thus, a stands for a number, and a number only, viz., 3.14159 but in the expression the angle TT,' that is, the angle 3.14159 ...,' there must be some unit understood. The unit understood here is a radian, and therefore the angle a stands for ' 1 radians' or 3.14159 ... radians, that is, two right angles.
Hence, when an angle is referred to, a is a very convenient abbreviation for two right angles.
So also the angle a or o' means ' a radians or 0 radians.'
The units in the three systems, when expressed in terms of one common standard, two right angles, stand thus :
1 The unit in the Sexagesimal Method = of 2 right angles.
1 « Centesimal
1 66 Circular
If D, G, and A denote the number of degrees, grades, and radians respectively in any angle, then D G 0
(1) 180 200 because each fraction is the ratio of the angle to two right angles.
9. Comparison of the Sexagesimal and Centesimal Measures of an Angle. — Although the centesimal method was never in general use among mathematicians, and is now totally abandoned everywhere, yet it still possesses some interest, as it shows the application of the decimal system to the measurement of angles. From (1) of Art. 8 we have
EXAMPLES. 1. Express 49° 15' 35" in centesimal measure.
First express the angle in degrees and decimals of a degree thus :
2. Express 87% 2 25" in degrees, etc. First express the angle in grades and decimals of a grade
Find the number of grades, minutes, and seconds in the following angles :
3.' 51° 4' 30". 4. 45° 33' 3". 5. 27° 15' 46".
Ans. 568 75' 0".
508 61' 20".37. 308 29' 19":75 ..., 1748 52' 12".962 ...,
6. 157° 4' 9".
Find the number of degrees, minutes, and seconds in the following angles :
7. 198 45' 95". 8. 1248 51 8". 9. 55° 18' 35".
Ans. 17° 30' 48".78.
111° 38' 44".592. 49° 39' 54".54.
10. Comparison of the Sexagesimal and Circular Measures of an Angle. From (1) of Art. 8 we have
D A 180