Question

The exact value of tan 5 π/12

Answer 1

C

Explanation:

Using the addition formula for tangent

tan(x + y) =
`(tanx+tany)/(1-tanxtany)` and the exact values

tan(
`(\pi )/(6)` ) =
`(1)/(√(3) )` and tan(
`(\pi )/(4)` ) = 1

tan(
`(5\pi )/(12)` ) = tan(
`(\pi )/(6)` +
`(\pi )/(4)` )

=
`(tan((\pi )/(6))+tan((\pi )/(4)) )/(1-tan((\pi )/(6))tan((\pi )/(4)) )`

=
`((1)/(√(3) )+1 )/(1-(1)/(√(3) ).1 )` ( multiply numerator/ denominator by
`√(3)` )

=
`(1+√(3) )/(√(3) -1)`

rationalise denominator by multiplying numerator/denominator by (
`√(3)` + 1)

=
`((1+√(3))(√(3)+1) )/((√(3)-1)(√(3)+1) )`

=
`(√(3)+1+3+√(3) )/(3-1)`

=
`(2√(3)+4 )/(2)`

=
`√(3)` + 2 → C