1. THE word Trigonometry in its primary sense signifies the measurement of triangles. From an early date the science also included the establishment of the relations which subsist between the sides, angles, and area of a triangle; but now it has a much wider scope and embraces all manner of geometrical and algebraical investigations carried on through the medium of certain quantities called trigonometrical ratios, which will be defined in Chap. II. In every branch of Higher Mathematics, whether Pure or Applied, a knowledge of Trigonometry is of the greatest value. B 2. Definition of Angle. Suppose that the straight line OP in the figure is capable of revolving abc it the point 0, and suppose that in this way it has passed successively from the position OA to the positions occupied by OB, OC, OD, .. then the angle between OA and any position such as OC is measured by the amount of revolution which the line OP has undergone in passing from its initial position OA into its final position OC. Moreover the line OP may make any number of complete revolutions through the original position OA before taking up its final position. H. K. E. T. 1 It will thus be seen that in Trigonometry angles are not restricted as in Euclid, but may be of any magnitude. The point is called the origin, and OA the initial line; the revolving line OP is known as the generating line or the radius vector. 3. Measurement of Angles. We must first select some fixed unit. The natural unit would be a right angle, but as in practice this is inconveniently large, two systems of measurement have been established, in each of which the unit is a certain fraction of a right angle. 4. Sexagesimal Measure. A right angle is divided into 90 equal parts called degrees, a degree into 60 equal parts called minutes, a minute into 60 equal parts called seconds. An angle is measured by stating the number of degrees, minutes, and seconds which it contains. For shortness, each of these three divisions, degrees, minutes, seconds, is denoted by a symbol; thus the angle which contains 53 degrees 37 minutes 2.53 seconds is expressed symbolically in the form 53° 37′ 2·53′′. 5. Centesimal Measure. A right angle is divided into 100 equal parts called grades, a grade into 100 equal parts called minutes, a minute into 100 equal parts called seconds. In this system the angle which contains 53 grades 37 minutes 2·53 seconds is expressed symbolically in the form 53′ 37' 2·53“. It will be noticed that different accents are used to denote sexagesimal and centesimal minutes and seconds; for though they have the same names, a centesimal minute and second are not the same as a sexagesimal minute and second. Thus a right angle contains 90 x 60 sexagesimal minutes, whereas it contains 100 x 100 centesimal minutes. Sexagesimal Measure is sometimes called the English System, and Centesimal Measure the French System. 6. In numerical calculations the sexagesimal measure is always used. The centesimal method was proposed at the time of the French Revolution as part of a general system of decimal measurement, but has never been adopted even in France, as it would have made necessary the alteration of Geographical, Nautical, Astronomical, and other tables prepared according to the sexagesimal method. Beyond giving a few examples in transformation from one system to the other which afford exercise in easy Arithmetic, we shall after this rarely allude to centesimal measure. In theoretical work it is convenient to use another method of measurement, where the unit is the angle subtended at the centre of a circle by an arc whose length is equal to the radius. This system is known as Circular or Radian Measure, and will be fully explained in Chapter VII. An angle is usually represented by a single letter, different letters A, B, C,..., a, ß, y,..., O, P, V,..., being used to distinguish different angles. For angles estimated in sexagesimal or centesimal measure these letters are used indifferently, but we shall always denote angles in circular measure by letters taken from the Greek alphabet. 7. If the number of degrees and grades contained in an angle D G 9 10° be D and G respectively, to prove that In sexagesimal measure, the given angle when expressed as the fraction of a right angle is denoted by D In centesimal 90* G 10 8. To pass from one system to the other it is advisable first to express the given angle in terms of a right angle. In centesimal measure any number of grades, minutes, and seconds may be immediately expressed as the decimal of a right angle. Thus 23 grades of a right angle='23 of a right angle ; 15 minutes angle; of a grade=·15 of a grade=0015 of a right ... 23° 15'2315 of a right angle. Similarly, 15 7 53·4" =·1507534 of a right angle. Conversely, any decimal of a right angle can be at once expressed in grades, minutes, and seconds. Thus 2173025 of a right angle=21·73025o =21 73.025' =218 73' 2·5". In practice the intermediate steps are omitted. OBS. In the Answers we shall express the angles to the nearest tenth of a second, so that the above result would be written 1°55′ 2.7". Example 2. Reduce 12° 13′ 14.3′′ to centesimal measure. This angle 13578487...ofaright angle =138 57' 84.9". 60) 14.3 seconds 60) 13-238333...minutes 13578487...of a right angle. 23. The sum of two angles is 805 and their difference is 18°; find the angles in degrees. 24. The number of degrees in a certain angle added to the number of grades in the angle is 152: what is the angle? 25. If the same angle contains in English measure x minutes, and in French measure y minutes, prove that 50x=27y. 26. If s and t respectively denote the numbers of sexagesimal and centesimal seconds in any angle, prove that 2508=81t. |