Question

Prove by vector method that sin(α + ß) = sin α cos ß + cos α sin ß

Answer 1

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