If 3 sin A + 5 cos A = 5, prove that (5 sin A - 3 cos A) = +-3

Asked by Kumarashish6345 | 22nd Feb, 2019, 08:57: AM

Expert Answer:

begin mathsize 16px style 3 space sin space straight A space plus space 5 space cos space straight A space equals space 5
Squaring space space on space space both space the space sides comma space we space get
left parenthesis 3 space sin space straight A space plus space 5 space cos space straight A right parenthesis squared space equals space 5 squared
rightwards double arrow 9 sin squared straight A space plus 25 cos squared straight A space plus space 30 sinAcosB equals 25 space space space space space space space space space space space space space space space space space space space space space space space space space space since space left parenthesis straight a space plus space straight b right parenthesis squared equals space straight a squared plus space 2 ab space plus space straight b squared
rightwards double arrow 9 left parenthesis space 1 space minus space cos squared straight A right parenthesis space space plus space 25 left parenthesis space 1 space minus space sin squared straight A right parenthesis plus space 30 sinAcosB space equals space 25
rightwards double arrow 9 minus space 9 cos squared straight A space plus space 25 minus 25 space sin squared straight A plus space 30 sinAcosB space equals space 25
rightwards double arrow 25 space sin squared straight A space minus space 30 sinAcosB plus space 9 cos squared straight A space equals space 9
rightwards double arrow open parentheses 5 sinθ space space minus space space 3 cosθ close parentheses squared space equals space 9 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space since space left parenthesis straight a space minus space straight b right parenthesis squared equals space straight a squared minus space 2 ab space plus space straight b squared
Taking space Square space root space on space both space the space sides comma space we space get
rightwards double arrow 5 sinθ space space minus space space 3 cosθ space space equals space plus-or-minus 3 end style

Answered by Yasmeen Khan | 22nd Feb, 2019, 10:54: AM