Question

Using vector method, prove that cos(α – β) = cos α cos β + sin α sin β

Answer 1

Let `bar"a" = bar"OA", bar"b" = bar"OB"`

Using vector method

Prove that cos(α – β) = cos α cos β + sin α sin β

Draw AL and BM perpendicular to the X axis, then

`bar"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"|"i"` = cos α `hat"i"`

`bar"LA" = sin alpha (+  hat"j")`

`baralpha = 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 so `bar"a"*bar"b" = |bar"a"*bar"b"|`

= `|bar"a"||bar"b"|` cos(α – ß) = cos(α – ß)  ........(3)

From (1) and (2)

`bar"a"*bar"b" = (cos alpha hat"i" + sinalpha hat"j")*(cos beta hat"i" + sin beta hat"j")`

= cos α cos ß + sin α sin ß  .......(4)

From (3) and (4)

cos(α – ß) = cos α cos ß + sin α sin ß