If a ` cos theta - b sin theta =c, " then" a sin theta + b cos theta` is equal to

Options

A. `+- sqrt(a^(2)+b^(2)+c^(2))`
B. `+-sqrt(a^(2)+b^(2)-c^(2))`
C. `+-sqrt(c^(2)-a^(2)-b^(2))`
D. None of these

Answer

B

Explanation

We have, ` a cos theta -b sin theta =c`
Squaring both sides, we get
` Rightarrow (a cos theta - b sin theta)^(2) =c^(2)`
`Rightarrow a^(2) cos^(2) theta +b^(2) sin^(2) theta-2ab sintheta cos theta=c^(2)`
` Rightarrow a^(2) (1 -sin^(2) theta) +b^(2) (1-cos^(2) theta) -2ab sin theta cos theta =c^(2)`
`Rightarrow a^(2) -a^(2) sin^(2) theta +b^(2) -b^(2) cos^(2) theta -2ab sin theta cos theta =c^(2)`
` Rightarrow a^(2) sin^(2) theta +b^(2) cos^(2) theta +2ab sin theta cos theta =a^(2) +b^(2) -c^(2)`
` Rightarrow (a sin theta + b cos theta)^(2) =a^(2) +b^(2)-c^(2)`
`Rightarrow a sin theta + b cos theta = +- sqrt(a^(2) +b^(2) -^(2))`