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Prot that theequation `k cos x-3s in x=k+1`possess a solution if`k in (-oo,4]dot`

Answer» Here, `kcosx-3sinx=k+1`, could be re-written as:
`k/sqrt(k^(2)+9) cosx -3/sqrt(k^(2)+9) sinx = (k+1)/sqrt(k^(2)+9)`
or `cos(x+phi) = (k+1)/sqrt(k^(2)+9)`, where `tanphi=3/k`
which possess a solution only is `-1 le(k+1)/sqrt(k^(2)+9) le1`
i.e., `|(k+1)/sqrt(k^(2)+9)| le1`
i.e., `(k+1)^(2)lek^(2)+9`
i.e., `k^(2)+2k+1 lek^(2)+9`
or `k le4`
The interval of k for which the equation `(kcosx-3sinx=k+1)` has a solution is `(-infty,4)`.


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