Elements of geometry and trigonometry . 4 * A T>» , and cot A= -^ , wc- liave tang Ax cot A — R-, a 7 • TTr sin a cos h. IK, or R. CB : CI BE cos h sm a IK: CB : CI :: CE : CK,or R : cos 6 : : cos a : CK- R cos a cos h. R The triangles DIL, CBE, having their sides perpendicular,each to each, are similar, and give the proportions, CB;DI : : CE : DL,or R : sin & : : cosa : DL^^^?^^ sin a sin h. CB : DI ; : BE : IL, or R : sin & : : sin a : IL= R But we have IK+DL=DFrrsin (a-Vh), and CK—IL=CF=cos (a+t). Hence . , , , V sin a cos ^ -f sin 6 cos asm {a-\-o) = cos (a+&) = R cos a cos I—sin a sin
Elements of geometry and trigonometry . 4 * A T>» , and cot A= -^ , wc- liave tang Ax cot A — R-, a 7 • TTr sin a cos h. IK, or R. CB : CI BE cos h sm a IK: CB : CI :: CE : CK,or R : cos 6 : : cos a : CK- R cos a cos h. R The triangles DIL, CBE, having their sides perpendicular,each to each, are similar, and give the proportions, CB;DI : : CE : DL,or R : sin & : : cosa : DL^^^?^^ sin a sin h. CB : DI ; : BE : IL, or R : sin & : : sin a : IL= R But we have IK+DL=DFrrsin (a-Vh), and CK—IL=CF=cos (a+t). Hence . , , , V sin a cos ^ -f sin 6 cos asm {a-\-o) = cos (a+&) = R cos a cos I—sin a sin h. R The values of sin {a—h) and of cos {a—h) might be easilydeduced from these two formulas ; but they may be founddirectly by the same figure. For, produce the sine DI till itmeets the circumference at M ; then we have BM=BD=&,and MI=ID==sin 6. Through the point M, draw MP perpen-dicular, and MN parallel to, AC : since MIrrDI, we have MN=IL, and IN=DL. But we have IK—IN=MP=sin (a—6),and CK+MN=CP=cos (a—6) ; hence PLANE TRIGONOMETRY. 225 . f ,x sin r7 cos h—sin /; cos asin {a—h)^ j^^ cos a cos
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Keywords: ., bo, bookcentury1800, booksubjectgeometry, booksubjecttrigonometry
