(20) (21) sin A. sin 2A + sin A. sin 44+ sin 24. sin 74 = =tan (0+$). (22) 2 coss A - 2 sin8 A = cos 24 (1+cos2 24). (23) (3 sin A-4 sin3 4)2+(4 cos3 A-3 cos A)2=1. =tan 54. cot (n-2) a-cot na (26) If tan a= and tan ẞ=1, prove tan (2a + B) = 1. *168. The following examples are symmetrical, and each involve more than two angles : Example 1. Prove that sin (a+B+7)=sin a. cosẞ.cos y+sinẞ.cos y. cos a + sin y. cos a. cos B- sin a. sin B. sinay. sin (a+B+y)=sin (a + B). cos y+cos (a +ẞ) sin ny + cos a. cos B. sin y- sin a. sin ẞ. sin y sin a. cos B. cos y + sin ẞ. cosy. cos a sin B. sin y. Q.E.D. .. sin a + sin ß + sin y − sin (a + ß+y) B-Y-2 cos 2a +B+y, 2 2a + B + y} = 2 sin B+Y COS 2 2 2 * EXAMPLES. Prove the following statements : (1) cos (a+B+y)= cos a. cos B. cos y - cosa. sin B. sin y (2) sin (a+ẞ-y)=sin a. cos B. cos y + sin ẞ. cosy.cos a sin B. - sin Y. cos a. cos ẞ+sin a sin ẞ sin y. (3) cos (a-B+y)= cos a. cos B. cos y + cos a. sin B. sin y - cos B. sin a. sin y+cos y . sin ß. sin a. (6) sin 2a + sin 2ẞ+ sin 2y - sin 2 (a+B+y) =4 sin (8+y). sin (y+a). sin (a+B). (8) sin (B+y-a) +sin (y+a− B) + sin (a + B − y) (9) sin (a + B+ y) + sin (B+y− a) + sin (y+a-ẞ) - sin (a +ẞ-y)=4 cos a. cos B. sin y. (10) cos x + cos y + cos z + cos(x+y+2) (11) cos 2x + cos 2y + cos 2z+ cos 2 (x + y + z) (12) (13) cos2x + cos2 y + cos2 z + cos2 (x + y + z) =2 {1 + cos (y + z). cos (z+x). cos (x+y)}. (14) sin2 x + sin2 y + sin2 z+sin2 (x + y + z) =2 {1 − cos (y + z). cos (z+x). cos (x+y)}. (15) cos2x+cos2y + cos2 z + cos2 (x + y − z) = -2 {1 + cos (x − z). cos (y-z). cos (x+y)}. (16) cosa.sin (6-y) + cos B. sin (y - a) + cos y . sin (a - ẞ)=0. (17) sin a. sin (B- y) + sin ẞ. sin (y - a) + sin y. sin (a-B)=0. (18) cos (a+B). cos (a − ß) + sin (3 + y) sin (8 − y) ⋅ cos (a+y). cos (a− y)=0. (19) cos (8-a). sin (8 − y) + cos (8 - B). sin (y - a) - cos (8-y). sin (B-a)=0. (20) 8 cos++++X.cos $+x-0 x+0-$ θ+φ-x 2 COS COS 2 2 2 = cos 20+ cos 2p+ cos 2x + 4 cos 6. cos. cos x + 1. CHAPTER XIII.** ON ANGLES UNLIMITED IN MAGNITUDE. II. 169. THE words of the proofs (on pages 118, 119) of the A, B' formulæ apply to angles of any magnitude. The figures will be different for angles of different magnitude. 170. The figure for the 'A-B' formulæ on page 119 is the same for all cases in which A and B are each less than 90°. The figure given below is for the proof of the ' A + B’ formulæ, when, A and B being each less than 90°, their sum is greater than 90°. N H M K R The words of the proof are precisely those of We may notice however that page 118. 171. Thus we have proved that the 'A, B' formulæ are true provided A and B each lie between 0° and 90°. The student can prove them for any other values by drawing the proper figure. The 'A, B' formulæ are therefore true for any values whatever of the angles A and B. 172. By the aid of the 'A, B' formulæ we can prove the formulæ of Art. 140. Example. Prove that sin (90° +4)=cos A. sin (90°+4)=sin 90o cos A+ cos 90° sin A, = 1 x cos 4+0 × sin A, cos A. Q. E. D. EXAMPLES. XLIV. Draw the figures for the first four of the following examples. (1) For the (A+B) formulæ, when A is greater than 90° and (A + B) less than 180o. (2) For the (A - B) formulæ, when A and B each lie between 90° and 180o. (3) For the (A+B) formulæ, when A lies between 90° and 180o, and (A + B) lies between 180o and 270o. (4) For the (4 – B) formulæ, when A lies between 180° and 270°, and (A - B) lies between 180° and A. Deduce the six following formulæ from the 'A, B' formulæ. (11) Assuming that the formula sin (A+B)=sin A. cos B +cos A. sin B is true for all values of A and B, deduce the rest of the 'A, B' formulæ by the aid of the results on p. 107. L. E. T. 10 |