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प्रश्न
Prove by vector method that sin(α + ß) = sin α cos ß + cos α sin ß
उत्तर
Let `bar"a" = bar"OA", bar"b" = bar"OB"` be the unit vectors making angles α and ß respectively with positive x-axis where A and B are as shown in the diagram
Draw AL and BM perpendicular to the X-axis, then
`"OL" = bar"OA"` = cos α
`|bar"OL"| = |bar"OA"|` cos α = cos α
`|bar"LA"| = |bar"OA"|` sin α = sin α
`|bar"OL"| = |bar"OL"| hat"j" = cos alpha hat"i"`
`bar"LA" = sin alpha(- hat"j")`
`bar"a" = bar"OA" = bar"OL" + bar"LA"`
= `cos alpha hat"i" + sin alpha hat"j"` ........(1)
Similarly `bar"b" = cos beta hat"i" - sin beta hat"j"` .......(2)
The angle between `bar"a"` and `bar"b"` is α + ß and the vectors `bar"b", bar"a", bar"k"` from a right handed system.
`bar"b" xx bar"a" = |bar"b"||bar"a"| sin(alpha + beta)hat"k"`
= `sin(alpha + beta)hat"k"` ........(1)
`bar"b" xx bar"a" = |(hat"i", bar"j", bar"k"),(cos beta, - sin beta, 0),(cos alpha, sin alpha, 0)|`
= `(sin alpha cos beta + cos alpha sin beta)hat"k"` .......(2)
From (1) and (2)
sin(α + ß) = sin α cos ß + cos α sin ß
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