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Question

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

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Solution

Given:6 cosθ+8 sinθ=96 cosθ=9-8 sinθ36cos2θ=(9-8 sinθ)2361-sin2θ=81+64sin2θ -144sinθ100sin2θ -144sinθ+45 =0Now, α and β are the roots of the given equation; therefore, cos α and cos β are the roots of the above equation.sinα sinβ=45100 (Product of roots of a quadratic equation ax2+bx+c=0 is ca.)Again, 6 cosθ+8 sinθ=98 sinθ=9-6 cosθ64sin2θ=(9-6 cosθ)264(1-cos2θ)=81+36cos2θ-108cos θ100cos2θ-108cosθ+17=0Now, α and β are the roots of the given equation; therefore, sinα and sinβ are the roots of the above equation.Therefore, cosα cosβ=17100Hence, cos(α+β)=cosα cosβ-sinα sinβ =17100-45100 =-28100 =-725

sin α+β =1-cos2α+β =1--7252 =576625 =2425

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