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: : 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 O L is the secant of a. Also the cosecant of a is the line O M. α 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, c: a :: 1: sin A, or sin A E α 1= c 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, Hence the magnitude of an angle is determined by the expression = arc radius Ex. Given the hypothenuse A B 11, of a right-angled triangle ABC, and the angle A 60, to find the other parts, a and b. If (fig. to Art. 2) AOB = 60 ̊, and OA = OB, then if the line A B be drawn, the triangle AO B will be equilateral; and hence OF = cos 60 † AO = ¦. Whence sin 60 = BF = √ (BO3 — O Fo) =√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, B F2 + F 02 = O B2, or sin 2A + cos 'A 1 cos 2A), and cos A = √(1-sin 2A). Also by the properties of similar triangles (Euc. vi., 4, 8), (1*). *The expression sin A 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. 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, EOQ, each equal to the angle A OB, and draw the perpendiculars P N, QR. Then (Euc. i. 26), the triangles, FO B, PNO, QR O, are equal, and FO = RO = PN, BF QR NO. Hence (Arts. 7, 11, 12), sin (A) = sin AOP = PN = FO = cos A. cos (T+A) = cos AOP = ON sin (-A) sin A OQ Q R = BF = cos (T- A) = = cos A O Q = OR= = Whence by the expressions (1)... (5), tan (+A)= - BF = - sin A - · OF= (11), cos A. (14). sin(+A) cos A cos (+ A) = sin ( 1 A) sin A Consequently the sine and cosecant of an angle are the same as those of its supplement; the cosine, tangent, cotangent, and secant, are equal in magnitude with contrary signs. = : The following are left as exercises for the student :cot (+A)= tan A.....(20), sec († τ+A) covers (+A) = 1 − cos A. . . . (22), vers (Ț — A) sec (π + A) = − sec A.....(24), tan (☛ + A) cot A.......(26), vers (π + A) = -sin A.....(28), cos (T+A) cot (+A) sin (+A) sec (2-A) = = = 1 + cos A. (23), sec A.......(30), vers (2π- A) = vers A. If, moreover, the angle AO B be positive when measured towards C, it will be negative when measured in the contrary direction (Art. 12). Hence the angle A O b will be denoted by A, and consequently by Arts. 7, 8, 11, 12, Sin (-A)-bF-BF-sin A. (32), cos(-A) OF= cos A..(33). And similarly for other functions of The numerical values of the trigonometrical functions of 30°, 60°, 45° may now be found. The values of other functions of 30, 45, and 60' may be found in a similar way. EXERCISES. 1. The tangent of an angle is ; find the surd expressions for the sine, cosine, secant, and cosecant of the same. Ans. If a be the angle, sin α = √√5, cos α = √√5, sec α = √5, cosec α = 5. 2. Find the complements of the angles 26 7' 8" 21, 98 16' 30", and 218 5' 6" 67. 3. Find the supplements of 155 7' 8" 9, and 224 5′ 8′′. 4. Find the circular measures of the angles 61, 72° 5' 20", and 88° 19' 30". Ans. 1·0646, 1·2582, 1·5415. 5. Find the angles whose circular measures are and Ans. 43 nearly, and 22.9. 6. Required the arc whose supplement is to its complement as four to one. Ans. O = 60. 7. Required the arc the sum of whose supplement and complement is to their difference as two to one. Prove the following relations :- Ans. 0 45. 8. Sina sin 28 + cos 2a cos 2ß + sin 1a cos 2ß + sin 2ß cos 2a = = 1. 10. Vers (+0) vers (π — →) + vers 0 vers (π — 0) = 1. 11. Seca tan'a (sec2ß − 1) (tan 3ß + 1) − sec 2ß tan 3ß (sec a x (tana + 1) = 0. 12. (Sin'a-sin'a sin B) (cosa - cosa cos 'B)-(sin B sina sin 28) cosa cos 8) = 0. FUNCTIONS OF TWO OR MORE ANGLES. 16. Expressions for the sine and cosine of the sum and difference of any two angles, and formulæ deducible from these. |
