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Vector Algebra

Question
CBSEENMA12034275
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Using vectors, prove that sin (A + B) = sin A + cos B + cos A sin B. 

Solution
Let OP and OQ be unit vectors making angles A and B with X-axis such that
angle space POQ space equals space straight A plus straight B
therefore space space space space space OP with rightwards arrow on top space equals space straight i with hat on top space cos space straight A space plus space straight j with hat on top space sin space straight A space space space space space space space space space OQ with rightwards arrow on top space equals space straight i with hat on top space cos space straight B space minus space straight j with hat on top space sin space straight A Now comma space OP with rightwards arrow on top space cross times space OQ with rightwards arrow on top space equals space left parenthesis 1 right parenthesis thin space left parenthesis 1 right parenthesis space sin space left parenthesis straight A plus straight B right parenthesis space left parenthesis negative straight k with hat on top space right parenthesis space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space therefore space space space space OP with rightwards arrow on top space cross times space OQ with rightwards arrow on top space equals space sin space left parenthesis straight A plus straight B right parenthesis space straight k with hat on top space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis Also comma space space space OP with rightwards arrow on top space cross times space OQ with rightwards arrow on top space equals space open vertical bar table row cell straight i with hat on top end cell cell straight j with hat on top end cell cell straight k with hat on top end cell row cosA sinA 0 row cosB cell negative sinB end cell 0 end table close vertical bar space equals space left parenthesis negative cos space straight A space sin space straight B space minus space sin space straight A space cos space straight B right parenthesis space straight k with hat on top therefore space space space space OP with rightwards arrow on top space cross times space OQ with rightwards arrow on top space equals space minus left parenthesis sin space straight A space cos space straight B space plus space cos space straight A space sin space straight B right parenthesis space straight k with hat on top space space space space space space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis

From (1) and (2), we get,
                        sin space left parenthesis straight A plus straight B right parenthesis space equals space sin space straight A space cos space straight B space plus space cos space straight A space sin space straight B.

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