Question

Cos ( α ) = √ 6/ 6 and sin ( β ) = √ 2/4 . Find tan ( α − β )

Answer 1

`\purple{ \bold{ \tan( \alpha - \beta ) = 1.00701798}}`

Explanation:

`\cos( \alpha ) = ( √(6) )/(6) = (1)/( √(6) ) \\ \\ \therefore \: \sin( \alpha ) = \sqrt{1 - { \cos}^(2) ( \alpha ) } \\ \\ = \sqrt{1 - \bigg( {(1)/( √(6) ) \bigg )}^(2) } \\ \\ = \sqrt{1 - {\frac{1}{ {6} }}} \\ \\ = \sqrt{ {\frac{6 - 1}{ {6} }}} \\ \\ \red{\sin( \alpha ) = \sqrt{ { \frac{5}{ {6} }}} } \\ \\ \tan( \alpha ) = (\sin( \alpha ) )/(\cos( \alpha ) ) = √(5) \\ \\ \sin( \beta ) = ( √(2) )/(4) \\ \\ \implies \: \cos( \beta ) = \sqrt{ (7)/(8) } \\ \\ \tan( \beta ) = (\sin( \beta ) )/(\cos( \beta ) ) = (1)/( √(7) ) \\ \\ \tan( \alpha - \beta ) = ( \tan \alpha - \tan \beta )/(1 + \tan \alpha . \tan \beta) \\ \\ = ( √(5) - (1)/( √(7) ) )/(1 + √(5) . (1)/( √(7) ) ) \\ \\ = ( √(35) - 1 )/( √(7) + √(5) ) \\ \\ \purple{ \bold{ \tan( \alpha - \beta ) = 1.00701798}}`