If a cos $\theta$ − b sin $\theta$ = c, then a sin $\theta$ + b cos $\theta$ =

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KollegeApply · Accepted Answer

Given: a cos$\theta$- b sin$\theta$=c Squaring on both sides, we have (a cos$\theta$-b sin$\theta$ ^2)=c^2 `$\Rightarrow$ a^2 cos^2 $\theta$+b^2 sin^2 $\theta$-2. a cos $\theta$. b sin$\theta$=c^2 `$\Rightarrow$a^2(1-sin ^2 $\theta$)+b^2(1-cos^2$\theta$)-2.a cos$\theta$. b sin $\theta$=c^2 `$\Rightarrow$a^2-a^2 sin^2$\theta$+b^2 cos^2 $\theta$-2.acos$\theta$. b sin$\theta$=c^2 `$\Rightarrow$-a^2 sin^2 $\theta$-b^2 cos^2 $\theta$-2 a cos$\theta$. b sin $\theta$=-a^2-b^2+c^2 `$\Rightarrow$-(a^2 sin^2 $\theta$+b^2 cos^2$\theta$+2.a cos$\theta$.b sin $\theta$)=-(a^2+b^2-c^2) `$\Rightarrow$a^2 sin^2 $\theta$+b^2 cos^2 $\theta$+2.a sin $\theta$.b 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-c^2`

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