find Solve the following five equations: 36. sin x + cos 2x = 1. 37. tan x=tan 4x. 38. sin 9x+ sin 5x + 2 sin2x=1. 39. sin 5x cos 3x=sin 9x cos 7x. 40. 41. cos ax cos bx=cos (a+c) x cos (b+c) x. If tan A=2, tan B=3, tan C=2+ √√3, 42. If tan A 44. If 2S=A+B+C, shew that cos3S+cos(S-A)+cos2(S-B) + cos(S-C) =2+2 cos A cos B cos C. 45. If vers A=x, vers B=bx, vers C=1-b, and A+B=C, find x. Demonstrate the following ten identities : 46. cos A+ cos B+cos (A+C)+cos (B+C) 48. sin (B-A) + sin2 (B-A+C) -2 sin (B-A) sin (B-A+C) cos C-sin2 C. 49. 4 cos 34 sin3 (60°-A)-4 sin 3A cos3 (60°-A) 50. cos A+cos (72°-A) + cos (72° +A)+cos (144o — A) +cos (144°+ A)=0. 51. 1+cos (A-B) + cos (B-C)+cos (C-A) 52. sin (A-B) + sin (B-C)+ sin (C-A) 53. sin 3 (A-15°) 54. =4 cos (A-45°) cos (A+15°) sin (A-15°). sin (2A-B-C) + sin (2B - A - C) 2 sin (B−A) sin (B–C) These formulæ serve to determine the sine and the cosine of half an angle, when the cosine of the angle is given. It will be seen that by reason of the double sign we have A 2 two values for sin and two values for cos ing to a given value of cos A. 199. The reason why there is more than one value for sin 4 and cos4, corresponding to a given value of cos A, is that corresponding to a given value of the cosine there is more than one value of the angle. Thus suppose that the angle COD has its cosine equal to cos A, then an angle equal to four right angles di minished by COD also has its cosine equal to cos A; this angle is denoted in the figure by the larger angle bounded by OC and OD'. If we take COD for A, then A is the angle COP, where OP is such that COP=POĎ. If we take for A the larger angle bounded by OC and OD', then 4 is the angle COP', where OP' is such that COP' = P'OD'. Also the angle measured in the negative direction between OC and OD', that is the angle COD' has its cosine equal to cos A. If we take COD for A, then A is the angle COP", where OP" is such that COP"=P"OD'. It is easy to see on examining the figure that COP' is the supplement of COP, and that OP' and OP" are in the same straight line. Hence it follows that the sines of COP, COP' and COP" are numerically equal but have not all the same sign; so also the cosines of COP, COP' and COP" are numerically equal but have not all the same sign. If any of the angles which we have taken for A be increased by any multiple of four right angles, we shall obtain an angle which has its cosine equal to cos A; it will be found on examining the figure that the sine and the cosine of half such an angle will coincide respectively with the sine and the cosine of one of the angles which we have already taken for A. A Thus we have accounted for the fact that sin and 2 A COS when expressed in terms of cos A have each two 2 values numerically equal but of opposite signs. 200. By assuming the result obtained in Art. 168 we can put the preceding explanation into a briefer form. All the angles which are comprised in the expression n 360° A, where n is any integer, positive or negative, have the same cosine as A. Hence we may expect that any formula which gives sin in terms of cos A will include A the sine of every angle which is comprised in the expression (n360o±A), that is in the expression n 180° ± A. Now sin (n 180° ± } A) = sin ( ± § A) if n be even, and-sin(+4) if n be odd. And thus, by Art. 159, we have the two values ±sin 1⁄2 A, and no more. Similarly we may expect that any formula which gives A COS in terms of cos A will include the cosine of every angle comprised in the expression n 180° ± § A. Now cos (n 180° ± 4) = cos (± 4) if n be even, and = cos(A) if n be odd. And thus, by Art. 159, we have the two values ±cos § A, and no more. 201. If in any case we actually know the value of A we know also the value of §4; and then we can settle which sign we ought to take in the formula for sin A, and which sign we ought to take in the formula for cos A. And even if we do not know the exact value of A we may know sufficient to enable us to make this selection; for example, if we know that A lies between 90° and 180o, then we know that A lies between 45° and 90°, and the positive sign must be taken in both the formulæ of Art. 198. 202. Remarks similar to those which have been made in the last three Articles will be found applicable also to numerous other results in Trigonometry in which the double sign occurs; for examples we may mention the remaining results of the present Chapter, or the result sin ▲ = ± √1 − cos2 A, and others of the same kind. We shall however not enlarge on this point, for we have given enough to illustrate the general principle; more applications will be found in Chapter VII. of the larger Trigonometry. |