If 2cos a + sine a=1 then
7 cos a + 6sin a=?
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If 2cos theta -sin theta =x …
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Class 10
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If 2cos theta -sin theta =x and cos theta - 3sin theta =y,prove that 2xsquare +y square -2xy =5
Posted by Puneet Kothari 1 year, 7 months ago
CBSE > Class 10 > Mathematics
1 answers
Sia 1 year, 7 months ago
Given, 2 cosθ - sinθ =x and
cosθ - 3 sinθ =y
Put the values of x and y in 2x2+y2−2xy (LHS) , we get
= 2(2 cosθ − sinθ)2 + (cosθ − 3 sinθ)2 − 2(2 cosθ − sinθ)(cosθ − 3 sinθ)
= 2(4cos2 θ− 4cosθ sinθ + sin2θ) + (cos2 θ− 6cosθ sinθ + 9sin2 θ) − 2(2cos2 θ− 7cosθ sinθ + 3sin2 θ)
= 8cos2θ − 8cosθ sinθ + 2sin2θ + cos2θ − 6cosθ sinθ + 9sin2θ − 2(2cos2 θ− 7cosθ sinθ + 3sin2 θ)
= 8cos2θ − 8cosθ sinθ + 2sin2θ + cos2θ − 6cosθ sinθ + 9sin2θ − 4cos2θ + 14cosθ sinθ − 6sin2θ
= 5cos2θ + 5sin2θ
= 5(cos2θ + sin2θ)
=5
= RHS
(2cosa)^2=(1-sina)^2
4cos^2a=1+sin^2a-2sina
4-4sin^2a=1+sin^2a-2sina
5sin^2a-2sina-3=0
5sin^2a-5sina+3sina-3=0
5sina(Sina-1)+3(sina-1)=0
(5sina+3)*(sina-1)=0
Sina=-3/5 or Sina=1
therefore for sina=-3/5 cosea=4/5
7cosa+6sina=7(-3)/5+6(3)/5=-3/5
for sina=1 and cosa=0
7cosa+6sina=6*1=6