Thus the formula has been extended to the case where one of the angles is obtuse and less than 180°. similar way the formula for cos(x+y) is extended to this case. By continuing this method both formulas are proved to be true for all positive values of x and y. Any negative angle y is equal to a positive angle y', minus some multiple of 360°. The functions of y are equal to those of y', and the functions of (x+y) are equal to those of (x+y'). 89 Therefore, the formulas being true for (x+y'), are true for (x+y). A repetition of this reasoning shows that the formulas are true when both angles, x and y, are negative. 33. Substituting the angle -y for y in formula (11), it becomes sin(x-y)=sin x cos (− y) + cos x sin (—y). But cos (−1) = cos y, and sin (—y)= — sin y. Therefore, sin (x − y) = sin x cos y − cosx sin y. Substituting (y) for y in formula (13), it becomes cos(x-y)= cos x cos (―y)—sin x sin (—y), = cos x cosy + sin x siny. Therefore, cos (x − y) = cos x cos y + sin x sin y.* 823 (12) (14) EXERCISES 34. (1.) Prove geometrically where x and y are acute and positive: sin(x−y)=sin x cosy— cosx siny, cos(xy)cos x cosy + sin r siny. * Formulas (12) and (14) are proved geometrically in § 34. The geometric proof is complicated by the fact that OD and DP are functions of -y, while the functions of y are what we use. Hint.—Angle AOQ=x, angle POQ=y, and angle AOP=(x−y). Then DP=sin(—y)=—siny; but DP is negative, therefore PD taken as positive is equal to sin y: OD=cos(-y)=cosy. Angle HPD-angle AOQ=x. their sides being perpendicular. sin(x-1)=SP=ED-PH. From right triangle OED, ED=(sin x) × OD=sin x cos y. Cos(x-1)=OS=OE+DH. From right triangle OÊD, OE=(cos x) × OD=cos x cosy. From right triangle DHP, DH=(sin x) × PD=sin x sin y. cos (x-1)=cos x cos y + sin x sin y. (2.) Find the sine and cosine of (45°+ x), (30°— x), (60°+ x), in terms of sin r and cos x. (3.) Given sin x=3, sin y=f3, x and y acute; find sin (x+y) and sin (x —y). (4.) Find the sine and cosine of 75° from the functions of 30° and 45°. Hint. 75°=(45°+30°). (5.) Find the sine and cosine of 15° from the functions of 30° and 45°. (6.) Given x and y, each in the second quadrant, sin x = = }, siny = 1; find sin(x+y) and cos (x —y). (7.) By means of the above formulas express the sine and cosine of (180° - x), (180°+x), (270°— x), (270°+x), in terms of sin x and cos r. (8.) Prove sin (60°+45°)+cos (60°+45°) = cos 45°. (9.) Given sin 45° = √√2, cos 45° = √2; find sin 90° and cos 90o. (10.) Prove that_sin (60° + x)— sin (60° — x) = sin x. TANGENT OF THE SUM AND DIFFERENCE OF TWO ANGLES 35. Tan(x+y)= = sin (r+y) _ sint cosy+cost sin y Dividing each term of both numerator and denominator of the right-hand side of this equation by cosa cosy, and In a similar way, dividing formula (12) by formula (14), we obtain tana - tany tan (x − y): 1+tanæ tany (16) FUNCTIONS OF TWICE AN ANGLE 36. An important special case of formulas (11), (13), and (15) is when y=x; we then obtain the functions of 2x in terms of the functions of x. From (II), sin(r+r)=sinx cosx+cosr sin r. FUNCTIONS OF HALF AN ANGLE 37. Equations (19) and (20) are true for any angle; therefore for the angle r. FORMULAS FOR SUMS AND DIFFERENCES OF FUNCTIONS 38. From formulas (11)–(14), we obtain Let then sin (x+y)—sin (x − y)=2 cosx siny; x=(u+v), y={(u —v). Substituting in the above equations, we obtain sinu + sinv = 2 sin√(u+v) cos} (u — v); sinu – sinv = 2 cos) (u +v) sin1⁄2 (u — v); (25) (26) cosu+cosv=2 cos1⁄2 (u+v) cos1⁄2 (u – v) ; (27) (28) 39. Express in terms of functions of x, by means of the formulas of this chapter, (5.) The sine and cosine of (45° —x); of (45° +x). (6.) Given tan 45° = 1, tan 30o =- § √3; find tan 75°; tan 15o. (9.) Prove cos (30°+y) — cos (30° — y) = — siny. (10.) Prove sin 3x= 3 sin x - 4 sin3x. Hint.-Sin 3x=sin(x+2x). (11.) Prove cos 3x=4 cos3 x 3 cos x. (12.) If x and y are acute and tan x=1, tan y=, prove that (x+y)=45°. (13.) Prove that tan (x+45°): = I + tan r (14.) Given siny and y acute; find siny, cosy, and tany. (15.) Given cos x = - and x in quadrant II; find sin 2x and COS 2.x. (16.) Given cos 45° = √2; find the functions of 2210. (17.) Given tan x = 2 and x acute; find tanr. (18.) Given cos 30° = √3; find the functions of 15o. (19.) Given cos 90° = 0; find the functions of 45°. (20.) Find sin 5r in terms of sin x. (21.) Find cos 5r in terms of cos x. (22.) Prove sin(x+y+z)=sin x cos y cos z+cos x sin y cos z+cos x cos y sin - sin x siny sin z. Hint.-Sin (x+y+z)=sin(x+y) cos z+cos(x+y) sin z. (23.) Given tan 2x=3 tan x; find x. (24.) Prove sin 32° + sin 28° = cos 2o. (25.) Prove tan x + cot x = 2 CSC 2x. (26.) Prove (sin x + cos (27.) Prove (sin x cos x)2 = 1 + sin x. sin x. |