The parts or degrees; each degree again into 60 equal parts called minutes; each minute into 60 equal parts called seconds; and so on. notation for degrees, minutes, etc., is °, ', '', etc., written similarly to indices in Algebra. Thus 21 degrees 2 minutes 17 seconds are written, 21° 2′ 17′′. 4. By the preceding Art. and Euc. vi. 33 (fig. to Art. 2), 90° 90° : <AOB:: quadrant AC: arc A B, or < AO B = ·A B. AC 90 Hence if (that is, 90° divided by the quadrant A C) be taken for AC the unit of angular measure, then the arc A B = < AOB. In this sense, therefore, the arc A B is the measure of the angle A OB, and either the one or the other is employed indifferently to express the inclination of the lines A O, BO. And, moreover, if the arc A B be given, the angle AOB can be found by this relation between the arc and the angle, and vice versa. Cor. The angle in a segment of a circle whose radius is the linear unit is measured by half the arc of the opposite segment. For as the arc A B or a measures the angle A O B, and as A OB (Euc. iii. 20) is double the angle in the segment B C D A, hence the angle in this segment is measured by a. 5. The semicircle ACE is represented by, and hence the quadrant AC by, and the whole circumference A CEDA by 2. Whence -a, and -a, are respectively the complement and supplement (Art. 2) of a. = It will be shown in a subsequent part of the Course, that = 3.14159265......, the value of this commonly used being = = 3.1416.* 6. Two right angles or 180° are represented by ; and therefore 90' by T. In many works of science, represents 31416, as well In this Treatise, two right angles will always be denoted by or 180, and 3·1416 by (the round letter), as in Art. 5. as 180. In the preceding articles it has been shown how angles can be expressed numerically by the subtending arcs of the circle whose radius is the linear unit. It will now be shown how arcs, and consequently their corresponding angles, can be measured by means of certain straight lines drawn in and about the circle, called the trigonometrical some eminent mathematicians to introduce the centesimal division, by which the right angle is divided into 100 equal parts or degrees, and each degree again into 100 equal parts or minutes, and so on. This method, though partially adopted in France, is not given in any English work. Its adoption in this country would have this advantage, that arithmetical operations could be performed on angles in the same manner as on any other decimal fractions; but it would require a complete transformation of all our tables which depend upon the sexagesimal division. *In the measurement of an angle by geometry, the right angle is assumed as the primitive angle, or angular unit, with which all other angles are compared. By the algebraic method, however, the angular unit (as in Art. 4) is 90° divided by the length of the quadrant to radius unity. Hence to convert the measure of an angle into degrees it is necessary to multiply it by this unit; thus, if it be required to find Sines functions of an arc. These lines are named since, tangents, etc., and they have been calculated (by methods which will be explained hereafter) for small divisions of the quadrant, By these functions of the angle or arc, it will be seen, we are enabled to compare the sides and angles of triangles. We will first describe these lines, and then deduce their geometrical properties. The lines BF, GA, referred to in the subsequent definitions, are drawn perpendicular to the diameter A E, and BK, CH, to CD (see fig. to Art. 2): 7. The sine of an arc, or of the angle of which the arc is the measure, is the perpendicular let fall from the extremity of the arc upon the diameter passing through the centre and the origin of the arc: thus BF is the sine of a, or of the angle A OB, and is written for brevity sin a. Cor. The chord of an arc is equal to twice the sine of half that arc. (By chord of an arc is meant the straight line passing through the origin and extremity of the arc.) For if we produce B F to meet the circle again in b, then (Euc. iii., 3, 28), the arc BA arc B Ab, and Bb = 2B F, or the chord of BAb 2 sin a. = = == 8. The tangent of an arc is the line drawn perpendicular to the diameter at the origin of the arc, and limited by the intersection of a line passing through the centre and the extremity of the arc: thus A G is the tangent of a, and is written tan a. 9. The secant of an arc is the line passing through or from the centre, through the extremity of the arc, and limited by the centre and the tangent: thus O G is the secant of a, and is written sec a. 10. The versed sine of an arc is the line or distance between the origin of the arc and the sine: thus AF is the versed sine of a, and is written vers α. 11. The cosine, cotangent, cosecant, and coversed sine, are the sine, tangent, secant, and versed sine of the complemental arc: thus BK, CH, OH, and CK are, respectively, the cosine, cotangent, cosecant, and coversed sine of a, and are written cos a, cot a, cosec a, covers a. Hence cot a = tan (w - α), cos α = sin - a), etc. Cor. Since cosα = BK OF, hence the cosine of an arc is equal to that part of the radius which is intercepted between the sine and the centre. == 12. The algebraical signs (+ or -) of the preceding functions of an arc (sin a, tan a, etc.) are determined by the geometry of coordinates. Thus the sine and tangent are positive or negative according as they are above or below the diameter A E, and the cosine and cotangent are positive or negative according as they are to the right or left of the diameter CD. Hence, The sine is positive (+) in the first and second quadrants, and negative (-) in the third and fourth. It *It must be kept constantly in mind that in this and the subsequent definitions, reference is made to a particular circle, viz., that whose radius is the linear unit. would not be correct to say that B F is the sine of the angle A O B to any radius. As will be seen presently, the sine in such case is expressed by a ratio. (See Functions of an Angle.) The tangent is positive in the first and third, and negative in the second and fourth, quadrants. The cosine is positive in the first and fourth, and negative in the second and third, quadrants. The cotangent is positive, in the first and third, and negative in the second and fourth, quadrants. The secant and cosecant, being revolving lines, are not drawn in the same direction, as any of the other functions, and hence their algebraic signs are not determined by the same principles.* They are, however, positive or negative according as they pass from, or through, the centre. Hence, The secant (like the cosine) is positive, in the first and fourth, and negative in the second and third, quadrants. The cosecant (like the sine) is positive in the first and second quadrants, and negative in the third and fourth. The versed sine and coversed sine are positive in all the quadrants, as they are always measured in the same direction: the former increases from 0 to 2 in the first and second quadrants, and then decreases from 2 to 0 in the third and fourth; the latter from 1 to 0 in the first quadrant, from 0 to 2 in the second and third, and from 2 to 1 in the fourth. We might proceed in a similar way to the fifth, sixth, etc., quadrants, but the functions would be merely repeated. 13. The following values are readily deduced from the preceding definitions : Angle. Sin. Cos. Tan. Cot. Sec. Cosec. Vers. Covers. The symbol denotes a quantity whose value is infinitely great. *The following definitions of the secant and cosecant, which were suggested to me by my friend and colleague, Mr. Heather, are not liable to this objection: : The secant of an arc is the straight line drawn from the centre through the origin of the arc, and limited by the centre and the line which touches the circle at the extremity of the arc. Thus OL is the secant of a. Also the cosecant of a is the line O M. E TRIGONOMETRICAL FUNCTIONS OF AN ANGLE. 14. Let A B C be a triangle, right-angled at C, and AD the radius or linear unit, as in the preceding definitions. Draw DE perpendicular to AC, and denote the sides of the triangle ABC opposite to the angles A, B, C, by a, b, c, respectively. Then by similar triangles, and Art. 7, ca: 1 sin A, or sin A a a a The ratios etc., are termed the trigonometrical functions of the angle BAC; they are obviously of the same value, whatever be the lengths of the lines A B, A C. These are the ratios to which reference is made in Art. 7, note. Cor. Let a be the measure of an angle as in Art. 4, and a the length of the corresponding arc to radius r; then, a 1:a::r: a, or a = -. r arc Hence the magnitude of an angle is determined by the expression radius Ex. Given the hypothenuse A B = 11, of a right-angled triangle A B C, and the angle A 60°, to find the other parts, a and b. If (fig. to Art. 2) AOB = 60, and OAO B, then if the line AB be drawn, the triangle A OB will be equilateral; and hence O F = cos 60°¦ AO = 1. Whence sin 60 BF = √ (B 02 – O F2) = 3. Consequently by the preceding formula (fig. of this Art.), FUNCTIONS OF ONE ARC OR ANGLE. 15. Relations amongst the trigonometrical functions of one arc or angle, and deductions from these. In the following investigations fig. 1 of Art. 2 is referred to, and the angle A O B is denoted by A. By the preceding definitions, and Euc. i., 47, BF2 + FO2 = O B3, or sin 2A + cos A = √(1-sin 2A). Also by the properties of similar triangles (Euc. vi., 4, 8), * The expression sin 2A denotes the second power of sin A. This power is sometimes written (sin A); and similarly for the second and higher powers of other functions. The former notation is adopted in this Treatise. OG2 = sin A : Hence also the following relations: OA2 + AG2, or sec 2A = 1 + tan 2A = 1 + (4), O B2 1= KO' (5), АО.ОС = GA (6). A F OA - OF, or vers A = 1 OC - OK, or covers A These properties are equally true for an angle greater than a right angle, Art. 12 being kept in mind. Scholium. It will be seen from the preceding relations that each of the trigonometrical functions of A can be expressed in terms of sin A; and similarly each might be expressed in terms of any other function. Hence we might have restricted ourselves to one function only in the definitions. This method, however, would have been much less effective in the complete development of trigonometry than that which has been adopted. The ancients, it would seem, confined themselves to one function (the chord), to which the Arabians added the sine. Again, at the points C and E make the angles COP, E O Q, each equal to the angle A OB, and draw the perpendiculars PN, QR. Then (Euc. i. 26), the triangles, FO B, |
