Elements of geometry and trigonometry . c or angle, we havecos Ar=sin (00^—A),cot A=rtang (<)0^—A),cosec A = sec (90^—A).The triangle MQC is, by construction, equal to the triangleCPM ; consequently CP—MQ : hence in the right-angled tri-angle CMl*, whose hy[)othemise is ecpjal to the radius, the twosides MP, CP are the sine and cosine of tin; arc AAf : hence,the cosine of an arc is copiai to tliat [)art of the radius inter-cepted between the centre and loot of the sine. The triangles CAT, CD8, are similar to the equal trianglesCPM, CQM ; lience they are similar to each other. Fromthese prin
Elements of geometry and trigonometry . c or angle, we havecos Ar=sin (00^—A),cot A=rtang (<)0^—A),cosec A = sec (90^—A).The triangle MQC is, by construction, equal to the triangleCPM ; consequently CP—MQ : hence in the right-angled tri-angle CMl*, whose hy[)othemise is ecpjal to the radius, the twosides MP, CP are the sine and cosine of tin; arc AAf : hence,the cosine of an arc is copiai to tliat [)art of the radius inter-cepted between the centre and loot of the sine. The triangles CAT, CD8, are similar to the equal trianglesCPM, CQM ; lience they are similar to each other. Fromthese principles, we shall very soon deduce the dilierent rela-tions which exist between the lir.(;s now defmed : before doing»o, however, we must examine the changers which those huesundergo, when the arc to winch they relate increases from zeroto 180. The angle ACD is railed {U\ first quadrant ; the angle DCB,the second nuadrant ; the; angle BCL, the third quadrant ; andthe angle LCA, liic fourth quadrant, s*27 216 PLANE VU. Suppose one extrem-iiy oi the arc remains fixed inA, vviiiie the other extremity,Jiiarked M, runs successivelythroughout the whole extentof the semicircuniierence,Irom A to B in the directionADB. When the point M is at A,or when the arc AM is zero,the three points T, M, P, areconfounded with the point A ;whence it appears that thesine and tangent of an arc zero, are zero, and the cosine and secant of this same arc, areeach equal to the radius. Hence if R represents the radius ofthe circle, we have sin 0=0, tang 0—0, cos 0==R, secO=R. VIÏI. As the point M advances towards D, the sine increases,and so likewise does the tangent and the secant ; but the cosine,the cotangent, and the cosecant, diminish. When the point M is at tlie middle of AD, or when the arcAM is 45^ in which case it is equal to its complement MD,the sine MP is equal to the cosine MQ or CP ; and the trian-gle CMP, having become isosceles, gives the proportionMP : CM : : 1 :
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