Question

If \[\alpha ,\beta \] are two different values of \[\theta \] lying between 0 and \[2\pi \] which satisfy the equation \[6\cos \theta +8\sin \theta =9\], find the value of \[\sin (\alpha +\beta )\].

Answer 1

Hint: Given that \[\alpha ,\beta \]are two different values of \[\theta \] in the interval 0 and \[2\pi \].The given equation is \[6\cos \theta +8\sin \theta =9\] .
For the given equation doing basic mathematical operations and from the trigonometric identities solving further leads to trigonometric properties and application of formula leads to the final answer.

Complete step-by-step answer:
Given \[6\cos \theta +8\sin \theta =9\]
\[6\cos \theta =9-8\sin \theta \]
Now squaring on both sides, we get
\[36{{\cos }^{2}}\theta =81+64{{\sin }^{2}}\theta -144\sin \theta \]
\[36(1-{{\sin }^{2}}\theta )=81-144\sin \theta +64{{\sin }^{2}}\theta \]
Solving further
\[100{{\sin }^{2}}\theta -144\sin \theta +45=0\]. . . . . . . . . . . . . . . . . . . . . . . (a)
The roots here are \[\sin \alpha ,\sin \beta \]. The product of roots is given by \[\dfrac{c}{a}\]
\[\sin \alpha \sin \beta =\dfrac{45}{100}\]. . . . . . . . . . . . . . (1)
Now again taking the equation\[6\cos \theta +8\sin \theta =9\]
\[8\sin \theta =9-6\cos \theta \]
Squaring on both sides,
\[64{{\sin }^{2}}\theta =81+36{{\cos }^{2}}\theta -108\cos \theta \]
\[100{{\cos }^{2}}\theta -108\cos \theta +17=0\]
The roots here are \[\cos \alpha ,\cos \beta \]. The product of roots is given by \[\dfrac{c}{a}\]
\[\cos \alpha \cos \beta =\dfrac{17}{100}\]. . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)
Substituting the values of (1) and (2)
Now \[\cos \alpha \cos \beta -\sin \alpha \sin \beta =\dfrac{17}{100}-\dfrac{45}{100}\]
\[\cos (\alpha +\beta )=\dfrac{-7}{25}\]. Then from the identity \[{{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1\]
We get the value of \[\sin (\alpha +\beta )=\dfrac{24}{25}\]

Note: The above equation (a) is in the form of a line \[ax+by+c=0\].Therefore the product of the roots is given as \[\dfrac{c}{a}\]. Care should be taken while doing the squaring because the identities come into picture and make the equation simpler.