Question

59. If 3 sin 0 + 5 cos 0 = 5, prove that 5 sino - 3 cos 0 = + 3.

JEE/Engineering Exams

Maths

Solution

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( (3 sin theta+5 cos theta)^{2}=5^{2} ) Squaring on both sides. ( (3 sin theta)^{2}+(5 cos theta)^{2}+2 times 3 sin theta 5 cos theta=25 ) ( left[a+b=a^{2}+b^{2}+2 a bright] ) ( 9 sin ^{2} theta+25 cos ^{2} theta+30 sin theta cos theta=25 ) ( 9left(1-cos ^{2} thetaright)+25left(1-sin ^{2} thetaright)+30 sin theta cos theta=25 ) ( left[sin ^{2} theta+cos ^{2} theta=1right] ) ( 9-9 cos ^{2} theta+25-25 sin ^{2} theta+30 sin theta cos theta=25 ) ( 9+25-left(9 cos ^{2} theta+25 sin ^{2} theta-30 sin theta cos thetaright)=25 ) ( 34-left(9 cos ^{2} theta+25 sin ^{2} theta-30 sin theta cos thetaright)=25 ) ( -left(25 sin ^{2} theta+9 cos ^{2} theta-30 sin theta cos thetaright)=25-34 ) ( left(25 sin ^{2} theta+9 cos ^{2} theta-30 sin theta cos thetaright)=9 ) ( (5 sin theta-3 cos theta)^{2}=9 ) ( (5 sin theta-3 cos theta)=sqrt{9} ) ( (5 sin theta-3 cos theta)=pm 3 ) ( mathrm{L.H.S}=mathrm{R.H.S} )