Pre Calculus

Question

Find the exact value of the expression tan(7pi/12)

a) 2(1+√3)^-2

b) 2(1-√3)^-2

c) -2(1+√3)^-2

d) -2(1-√3)^-2

Answer 1

We can attempt to write the angle as a sum or difference of angles.  We use angles we can easily find tangent of.

 

tan(4π/12 + 3π/12) =

 

tan(π/3 + π/4)

 

 

Now we can use the addition angle identity for tangent:

 

tan(x + y) = (tanx + tany) / (1 - tanxtany)

 

 

Therefore,

 

tan(π/3 + π/4) = [tan(π/3) + tan(π/4)] / [1 - tan(π/3)tan(π/4)]

 

                     

 

Use the fact that

 

tan(π/3) = sin(π/3) / cos(π/3)

             = (√(3) / 2) / (1 / 2)

             = √(3)

 

 

tan(π/4) = sin(π/4) / cos(π/4)

             = (√(2) / 2) / (√(2) / 2)

             = 1

 

 

Finish up here.