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§ 1. THE TRIGONOMETRICAL ANGLE. 1. In ordinary Geometry, an angle is entirely determined by the relative position of its two bounding straight lines. Thus, it is represented in a figure by two straight lines meeting at a point.





Two such angles are said to be equal if the pair of linessupposed to be rigidly connected—which bound one angle may be superposed exactly on the pair of lines which bound the other angle. Thus, in the accompanying figure, the acute angles A and B are equal; and the obtuse angles C and D are equal.

2. But in Trigonometry we consider the mode in which an angle is described.

An angle is described by the revolution of a straight line about one of its extremities which is fixed.

Thus, the hand of a clock describes an angle by revolving about the centre of the clock-face.

The lines which bound a geometrical angle are those between which the revolver moves in describing the corresponding trigonometrical angle.


J. T.


The trigonometrical angle is measured by the Amount of Revolution which the revolver undergoes in its movement between the two bounding lines.

3. A line which has revolved continuously from any position back to that same position is said to have described a Complete Revolution.

Thus, the minute-hand of a clock describes a complete revolution in an hour : the hour-hand in 12 hours.

The quarter of a complete revolution is called a quadrant. This angle is the same as the geometrical right-angle. In describing a complete revolution, the Quadrants are called the first, second, third, and fourth, respectively, in the order in which the revolver passes through them.

4. To every Geometrical angle (see Fig. of Art. 1) there corresponds an indefinite number of Trigonometrical angles. For there is an indefinite number of ways by which the revolver may supposed to have moved from the one to the other of the two lines which bound the Geometrical angle.

(1) The revolver may begin to revolve towards either side of one of the bounding lines to reach the other.

(2) The revolver may describe any number of complete. revolutions before coming to rest.

Thus, to represent in a Figure the Trigonometrical angle, we must indicate the side of the bounding lines on which the


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angle is formed, and the number of complete revolutions which have been described.

A part of a small spiral terminated by the two bounding lines may be used to indicate the Amount of Revolution.

Thus, the four acute angles in the accompanying figure are geometrically equal, but the spirals indicate in each case different amounts of revolution.

Similarly as regards the four obtuse angles.

The position of the minute-hand of a clock gives us the geometrical angle between its present position and its position at the beginning of the hour. But it is the position of the hour-hand which tells us how many complete revolutions the minute-hand has described since 12 o'clock. Thus the minute-hand and the hour-hand combined give the measure of the trigonometrical angle which the minute-hand has described. The way of revolution is of course indicated by the order of the numbers on the circumference of the clock-face.

5. Sometimes it is necessary to distinguish between the line from which the revolver starts and the line at which the revolver stops in describing any angle. The first is called the Initial Line and the second the Final Line. The words initial and final thus refer to the beginning and end, respectively, of any particular revolution considered. The letters I and F will be used to indicate the initial and final lines bounding any angle. 6. From the above it follows that, whereas every

Geometrical angle is [properly) less than two right-angles, a Trigonometrical angle may have any magnitude whatever. In fact, any Trigonometrical angle greater than two right-angles may be regarded as made up by the summation of a number of Geometrical angles each less than two right-angles. [Such an angle is contemplated in Euc. VI. 33.] And

Two trigonometrical angles are said to be equal, if the geometrical angles into which the one may be divided are equal, each to each, to those into which the other may be divided.

Subdivisions of the Right-angle. 7. The right-angle is subdivided in two ways according to the following tables.

Sexagesimal method.
1 Right-angle = 90 degrees, written 90°
1° i.e. 1 Degree = 60 minutes, written 60'
1' i.e. 1 Minute = 60 seconds, written 60"

Centesimal method.


1 Right-angle = 100 grades, written 100%
18 i.e. 1 Grade = 100 minutes, written 100
l'i.e. 1 Minute = 100 seconds, written 100".



8. The advantage of the centesimal method is that it is a mere application of the decimal system of notation. Thus 2 rt. 2 s 358 27' 25":4 = 2 rt. 2s 358 27'254

= 2 rt. ZS 358.27254

= 2.3527254 of a rt. L. Similarly 98 3' 1".73 = .09030173 of a rt. i.

In reducing angles so expressed to decimals of a right-angle, we have only to transform the integral numbers into decimals, taking care to express the units of each kind as hundredths of the next higher subdivision.

9. Observe that a complete revolution = 4 right angles = 360°, and that half a complete revolution = 2 right angles = 180o.

The student must be careful to avoid confusing the three different uses of the words minute and second.

The English or sexagesimal angular minute is a sixtieth of an angular degree.

The French or centesimal angular minute is a hundredth of an angular grade.

The minute of Time is a sixtieth of an hour.
Similarly as regards the second.


10. To reduce an angle from the Sexagesimal to the Centesimal measurement.

Reduce the angle to the decimal of a right-angle, and mark off the grades &c. as in Art. 8.


17° 11' 22".98.
60 ) 22.98 seconds
60 ) 11.383 minutes
90 ) 17.189716 degrees

190996851 right-angle

198 996":850.


11. To reduce an angle from the Centesimal to the Sexagesimal measurement.

Mark off the angle as a decimal of a right-angle as in Art. 8, and reduce to degrees &c. Ex.

198 996".851
•190996851(851 &c.) right-angle

17:189716(66659 &c.) degrees

11:3829(96 &c.) = 11383 minutes


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12. In all tables of reference, records of observation &c. angles are expressed in terms of the sexagesimal subdivisions of the right-angle. The immense labour which would be involved in reducing these to the centesimal system has not yet been undertaken. Hence the centesimal subdivisions, though introduced by the French at the beginning of the century, have performed no service except that of supplying elementary exercises in Reduction for the Trigonometrical student.

13. Angles which together make up one right-angle are said to be Complementary to one another: and each angle is called the Complement of the other.

Thus, the pairs of angles 30° and 60°, 47° 13' 29" and 42° 46' 31" are complementary pairs. If A denote any angle, 90° -- A denotes its complement.

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