II. When the angle BAC exceeds a right angle (fig. 2.), PN diminishes and AN increases; therefore the absolute values of the sine, tangent, and secant diminish, whilst those of the cosine, cotangent, and cosecant, increase. Also since the value of PN is positive and that of AN negative, the values of the sine and cosecant are positive, and those of the rest of the Trigonometrical Ratios negative. When the angle increases up to π, so that PN vanishes, and AN-r, we have III. Similarly, as the angle BAC (fig. 12.) increases from π to the sine, tangent, and secant increase, and the 3 п 2 other Trigonometrical Ratios diminish; and as the values both of PN and AN are negative, the sine, cosine, cosecant, and secant are negative, but the tangent and cotangent positive; and when the angle increases up to 3п so that AN vanishes, 2 IV. Lastly, as the angle BAC (fig. 13.) increases from 2 to 2π, the sine, tangent, and secant diminish, and the rest increase; and as the value of PN is negative, but that of AN positive, the cosine and secant are positive, and the rest negative; and when the angle becomes 2π, we have 27. In the applications of Analysis, as has been stated, we frequently have to consider angles which contain several π times: we must therefore give formulæ for expressing the Trigonometrical Ratios of all such angles by those of other angles less than . We shall especially consider the sine and 2 cosine, which are the Ratios most used; and as every angle greater than must consist of an angle <, together with once or several times repeated, we shall first examine what would be the sine and cosine of π + 0, 0 being less than π. Let the angle BAC (fig. 14.) be denoted by 0; produce CA to C', then the positive angle BAC' will be denoted by π+0; PN P'N' and these angles will have equal sines AP' AP ; but as the lines PN, P'N' have contrary positions, they must be affected with different algebraic signs; the cosines in like manner AN AN' are equal and must be affected with contrary AP' AP' algebraic signs; Next suppose increased by 2π, then the line AC will return to its original position, and all the Trigonometrical Ratios will remain the same; ... sin (2π + 0) = sin 0, cos (2π + 0) = cos 0. In general, whatever be the magnitude of the angle 0, if we add to it π, or any odd multiple of, the bounding radius will be transported to a position exactly opposite to that which it first occupied, and then, as is evident, the sine and cosine have only their algebraic signs altered, but not their magnitudes; but if we add to it 2π, or any even multiple of π, the bounding radius returns to its original position, and the Trigonometrical Ratios are altered neither, in magnitude nor algebraic sign. Hence the sine of any multiple of π will be zero; and the cosine will equal + 1 or 1 according as the multiple is even or odd; i. e. sin nπ = 0, cos nπ = ( − 1)". From the preceding results we readily perceive that tan (π + 0) = tan 0, tan (2π + 0) = tan 0. 28. It is proper to observe, that the formulæ proved here, and at Arts. 23, 24, are true for positive and negative angles of all magnitudes. The first were proved only for values of 0 between zero and π; changing into π + 0, they become sin (-) = sin (π + 0), cos (0) = cos (π + 0), which are evidently true by (2) and (4). We may now again increase by, and so on to any extent; also putting - 0 instead of 0, we see that the two formule are still true; therefore they hold for all angles whatever. Formula (2) we have seen to be true for all positive angles; also if we replace by -0, they become identical with (1), therefore they subsist also for negative angles. In formulæ (4) it is evident that may be replaced by 0; and from the way in which these formulæ are established, there is no limitation to the magnitude of 0. Formula (3) we have seen to be true for all positive values of 0; and since the addition of 2π to any angle, positive or negative, makes no alteration in its sine or cosine, they must be true for negative values of 0. Hence also, from Art. 12, we have cos (~ + 0) = sin ( − 0) — — sin 0, (+ = 29. It is now easy to reduce the Trigonometrical Ratios of any angle whatever, to those of an angle less than 90o. We must first suppress 360° as often as we can, and the sine, cosine, and tangent remain unaltered. We must next suppress 180° (if the angle exceed 180°) and change the signs of sine and cosine, but not of tangent. If the angle which now remains be greater than 90°, we must take its supplement, and change the algebraic signs of the cosine and tangent, but not of the sine. The values of the other Trigonometrical Ratios may be found by expressing them by the sine, cosine, or tangent. = = cos 129° = tan 135°-tan 45o. sin (-1029°) sin (- 309") = - sin (- 129°) = sin 51°. = 30. Since sin sin (π − 0) = − sin (!— 0) = − sin (π + 0), and since we are at liberty to add or subtract any multiple of 2π to or from an angle without altering the values of its Trigonometrical Ratios, we have sin 0 = sin (2n + 0) = sin {(2n ̧+ 1) π - 0} = − sin (2nπ- 0) = − sin { (2n + 1) π + 0} ; or, expressed by a single formula, sin 0 = sin {nπ + ( − 1)" 0}. or, expressed by a single formula, cos 0=(−1)" cos (nπ±0). And since tan = − tan (π−0) = − tan ( − 0) = tan (π + 0), = tan 0=tan (2noπ + 0) = − tan { (2n+1) π − 0} = −tan (2nπ-0) = tan {(2n + 1) π + 0 } ; or, expressed by a single formula, tan ✪ = tan (~π + 0). In all these expressions n is zero or any positive or negative integer; and is any angle positive or negative. Similar formulæ may of course be obtained for the other Trigonometrical Ratios. On the angles which correspond to given values of the sine, cosine, &c. 31. The preceding results give occasion for the important remark that there exists an infinite number of angles which have the same Trigonometrical Ratios. We will now therefore suppose the values of some of those ratios to be given, and determine all the angles which correspond to them. To find all the values of the angle 0 which satisfy the equation sin = ɑ. Constract, as in Art. 14, the angle BAC = a, (fig. 9.), having its sine = a; and let BAC-a be the supplement of BAC. Then the equation sin = a can be satisfied only by angles which are bounded by AB and AC, or by AB and AC; hence the positive angles will be BAC, and BAC increased by any multiple of 2π; and BAC, and BAC increased by any multiple of 2; i. e. they will be 2nπ + a, and 2nπ + (π − a); and the negative angles will be BAC, BAC, reckoned in the negative order, and the sums of each of these angles and any multiple of 2π taken negatively, i. e. they will be or - − 2nñ — (2π − a), and − 2nπ – (π + a), 2 (n + 1) π + a, and (2n+1)π-a; both of which series of angles are comprised in the expression n being any positive or negative integer not excluding zero, which, consequently, is the general value of 0. 32. To find all the values of the angle which satisfy the equation cos 0 = a. Construct the angle BAC = a (fig. 10.) having its cosine equal to a, and make the negative angle BAC = = BAC. Then the equation cosa can be satisfied only by angles which are bounded by AB and AC, or by AB and AC. Hence the positive angles will be BAC, and BAC increased by any multiple of 2; and BAC' (reckoned in the positive order), and BAC' increased by any multiple of 2; i. e. they will be 2n+a, and 2nπ + (2 x − a); and the negative angles will be BAC, BAC, reckoned in the negative order, and the sums of each of these angles and any multiple of 2π taken negatively, i. e. they will be both of which series of angles are comprised in the expression 2nra, n being any positive or negative integer not excluding zero, which, consequently, is the general value of 0. |