. Elements of plane and spherical trigonometry . #,or sin 2x = 2 sin x cos a;. (54) In the same manner, from (50) and (52), cos 2x = cos2 x — sin2 x, (55) tan 2x = 2 tan x . (56) 1 _ tan2 x K J 39. Functions of the Half-Angle. Formula (55) expresses the fact that the square of thecosine of any angle, diminished by the square of the sine ofthe angle, is equal to the cosine of twice the angle. We maytherefore write it thus: c d 5 50 PLANE TRIGONOMETKY. cos2 \x — sin2 \x = cos \x -\- sin2 ix= 1. Also from (20), Subtracting (a) from (/?), 2 sin2 §x=l — cos x, or sin ?—V1 COS # Adding (a) to
. Elements of plane and spherical trigonometry . #,or sin 2x = 2 sin x cos a;. (54) In the same manner, from (50) and (52), cos 2x = cos2 x — sin2 x, (55) tan 2x = 2 tan x . (56) 1 _ tan2 x K J 39. Functions of the Half-Angle. Formula (55) expresses the fact that the square of thecosine of any angle, diminished by the square of the sine ofthe angle, is equal to the cosine of twice the angle. We maytherefore write it thus: c d 5 50 PLANE TRIGONOMETKY. cos2 \x — sin2 \x = cos \x -\- sin2 ix= 1. Also from (20), Subtracting (a) from (/?), 2 sin2 §x=l — cos x, or sin ?—V1 COS # Adding (a) to (/S)or 2 cos2 Jx = 1 -f cos x, COS *# = /1 + cos JgDividing (57) by (58), tan2 \x or tan \x 1 — cos x 1 -f- cos x QO&X V 1 -f- cos # (a) (57)(58) (59) 40. Sum and Difference ofthe Sine and Cosine of TwoAngles. Eeferring to Fig. 17, let AOGx= 0 and AOC=<p. Then AOB= i(f+0) and BOC= IQp — 0).We then have FE Fig. 17 {bis).,0 sm — sin (9 =. HEXOE o ~ OE ~^~ OE ~ OE i^ff HE, _FE — HEr OE OE OE FE = GD if- jOJ, ^^1= GD — ID = GD — KE, FE+HEX = 2GD, FE—HE=2KE. sin 2> -j- sin 0 = 2 sin $p — sin 0 GD OEKEOE 2DG^ GD OB OE2 KE EDED A OE GENERAL FORMULAE. 51 But Z— = sin A OB = sin I O + 0), °P- = cos BOC = cos | (o — 0),^- = cos D^ = cos A OB = cos \ U + 0). and -— = sin BOC= sin g (cr — 0). sin cr -f sin 0 = 2 sin J (e> -f 0) cos | (cr — 0),and sin r — bid 0 = 2 cos I (c -j- 0) Bin I (jp — 0). Again, we have OF A n OH OH cos cr = . and cos 0 = —— = ——, (XE 0E1 OE a OF+OH OH-{-OF COS tp — COS 0 = and cos <s — cos 6 OE OE OF— OH OH— OF OE OE But 0H= OG + GH = OG + PG, and OF=OG — FG. 0H+ 0F=20G,and 0H—0F=2FG. Hence cos *, + cos 0 = 2-^ = 2^- X -£?» * Q# OB OE and cos w — cos 0 = —2 —— = —2 X ^r^r- OE BE OE But ^^ = cos A OB = cos * O -f 0), OB - v- -r y, ^ = «»2*Oa=coB *(, — *), — = — = sin BEK= sin ^ OD = sin h UBE BE v and ^^ = sin B0C= sin | (c? — 6).
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