If α,β are two different values of x lying between 0 And 2π which satisfy the equation 6 cos x + 8 sin x = 9, find the value of Sin(α+β).

Given 6 cos x + 8 sin x = 9


Case 1:


⇒ 6 cos x = 9 – 8 sin x


Squaring on both sides,


⇒ 36 cos2 x =(9 – 8 sin x)2


We know that cos2 x = 1 – sin2 x.


⇒ 36(1 – sin2 x) = 81 + 64 sin2 x – 144 sin x


⇒ 100 sin2 x – 144 sin x + 45 = 0


∴ cos α And cos β are the roots of the a bove equation


⇒ sin α sin β = 45/100


Case 2:


⇒ 8 sin x = 9 – 6 cos x


Squaring on both sides,


⇒ 64 sin2 x =(9 – 6 cos x)2


We know that sin2 x = 1 – cos2 x


⇒ 64(1 – cos2 x) = 81 + 36 cos2 x – 108 cos x


⇒ 100 cos2 x – 108 cos x + 17 = 0


∴ sin α And sin β are the roots of theAbove equation


⇒ cos α cos β = 17/100


Consider cos(α + β),


We know that cos(A +B) = cosA cosB - sinA sinB



We know that






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