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ocos y - sin x sin y3cos +y)Consider the uncicile withthe angle POP Then (x+3)centre at the origin. Let.x be the an+1) is the angle POP.. Also let (-y) be the:Therefore, P. P P, and P, wР.leos cr +y, sin (x +3% P,Icos (-у), sin (-У)] and P. (1. 0)(Fire:P. P, and P, will have the coordinates P |
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Answer» Draw first the two anglesaaandbbin the trigonometric circle. ProjectAEAEonADAD. Since|AE|=1|AE|=1and that the triangleADEADEis rectangle inDD, you have that|AD|=cos(b)|AD|=cos(b)and|ED|=sin(b)|ED|=sin(b). Now projectADADonACAC, since the triangleACDACDis rectangle inCCand that|AD|=cos(b)|AD|=cos(b), you have that|CD|=cos(b)sin(a)|CD|=cos(b)sin(a)and|AC|=cos(a)cos(b)|AC|=cos(a)cos(b). Finally projectEDEDonEFEF. The angleFEA^=aFEA^=a. Moreover, the triangleEFDEFDis rectangle onFFand thus|EF|=cos(a)sin(b)|EF|=cos(a)sin(b)and|FD|=sin(a)sin(b)|FD|=sin(a)sin(b). Now, just remark that|FB|=|CD||FB|=|CD|and|BC|=|FD||BC|=|FD|and you get sin(a+b)=|EF|+|FB|=cos(a)sin(b)+sin(a)cos(b)sin(a+b)=|EF|+|FB|=cos(a)sin(b)+sin(a)cos(b) and cos(a+b)=|AC|−|BC|=cos(a)cos(b)−sin(a)sin(b) thanks |
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