Question

Find the exact value of 0 tan165 using the appropriate identity formula. Rationalize denominator if needed.

Answer 1

- sqrt(7-4sqrt(3)) = -0.26794

tan165º = -0.26794

Explanation:

On EDG

Answer 2

`tan165^0 = -2 + √(3)`

Explanation:

Given;

tan165⁰

165⁰ is in the second quadrant, and sine is positive in this quadrant, thus tan will be negative.

In second quadrant, tanθ = -tan(180 - θ)

tan165⁰ = -tan(180-165)

tan165⁰ = -tan15⁰

`- tan15^0 = -tan(60 - 45)\\\\ tan(\alpha - \beta) = (tan \alpha -\ tan \beta)/(1 + tan \alpha tan \beta)\\\\ -tan(60 - 45) = -((tan 60 - tan 45)/(1+tan60tan45) )\\\\ -((tan 60 - tan 45)/(1+tan60tan45) )= -((√(3) \ - 1)/(1+ \ √(3) ) ) = (1 \ -√(3) )/(1+ \ √(3) )\\\\(1 \ -√(3) )/(1+ \ √(3) ) =( (1 \ -√(3) )/(1+ \ √(3) ))((1 - \ √(3))/(1 - \ √(3)) ) = (1-√(3) -√(3) \ +\ 3 )/(1\ -\ 3) = (4-2√(3) )/(-2)`

`(4-2√(3) )/(-2) = -2 + √(3)`

Therefore,
`tan165^0 = -2 + √(3)`