Q. 504.4( 7 Votes )

If 4 cos θ + 3 sin θ = 5, find the value of tan θ.

Answer :

Given : 4 cos θ+ 3 sin θ = 5

Squaring both the sides, we get


(4 cos θ+ 3 sin θ)2 = 25


16 cos2 θ + 9 sin2 θ + 2(4cos θ)(3sin θ)= 25 [ (a + b)2 =a2 +b2 +2ab]


16 cos2 θ + 9 sin2 θ + 24 cosθ sinθ = 25


Divide by cos2 θ, we get



16 + 9tan2 θ + 24 tanθ = 25sec2 θ


16 + 9tan2 θ + 24 tanθ = 25(1 + tan2 θ) [ 1+ tan2θ = sec2 θ]


16 + 9tan2 θ + 24 tanθ = 25+ 25 tan2 θ


16tan2 θ – 24tanθ + 9 = 0


16tan2 θ – 12 tanθ – 12 tanθ +9 = 0


4tanθ (4tan θ 3) 3(4tan θ 3) = 0


(4tan θ 3)2 = 0



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