1.

Find the range of the function `5sinx -12cosx +7`

Answer» `f(x)=5sinx-12cosx+7`
`sqrt(a^2+b^2)/sqrt(a^2+b^2)(asinx++bcosx)`
`sqrt(a^2+b^2)(a/sqrt(a^2+b^2)sinx+b/sqrt(a^2+b^2)cosx)`
`sinalpha=a/sqrt(a^2+b^2)`
`cosalpha=b/sqrt(a^2+b^2)`
`sqrt(5^2+12^2)=sqrt(25+144)=sqrt169=13`
`13/13(5sinx-12cosx)+7`
`13(5/13sinx-12/13cosx)+7`
`cosalpha=5/13`
`sinalpha=12/13`
`13(cosalphasinx-sinalphacosx)+7`
`13(sin(x-alpha))+7`
`-1<=sin(x-alpha)<=1`
`-13<=13sin(x-alpha)<=13`
`-13+7<=13sin(x-alpha)+7<=13+7`
`-6<=f(x)<=20`
`f(x) in [-6,20]`.


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