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Question
If`( 2 sin theta + 3 cos theta) =2 , " prove that " (3 sin theta - 2 cos theta) = +- 3.`
Solution
Given ,`(2 sin theta + 3 cos theta ) = 2 .....(i)`
We have `( 2 sintheta + 3 cos theta )^2 + ( 3 sin theta - 2 cos theta )^2`
=` 4 sin^2 theta + 9 cos^2 theta + 12 sin theta cos theta + 9 sin^2 theta + 4 cos^2 theta - 12 sin theta cos theta`
=`4 ( sin^2 theta + cos^2 theta ) + 9 ( sin^2 theta + cos^2 theta )`
=`4+9`
=13
i.e .,`( 2 sin theta + 3 cos theta ) ^2 + ( 3 sin theta - 2cos theta )^2 = 13`
= > `2^2 + (3 sintheta - 2 cos theta )^2 = 13`
= > `( 3 sin theta - 2 cos theta ) ^2 = 13-4`
= > `( 3 sin theta - 2 cos theta ) ^2 = 9 `
= > `( 3 sin theta - 2 cos theta ) = +- 3`
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