Find tan ( u/2 ) if cot u = 1.878

Question

and u is in Quadrant-III.

what is tan(u/2)=?

Answer 1

The tan 2a formula is tan 2x = 2 tan x/(1 - tan² x)

We can then solve for the half-angle formula

2 x/(1 - x^2) = y

2x = y- x^2y

x^2y + 2x - y = 0

x = (-1 ± √(1 + y²))/y

tan x/2 = (-1 ± √(1 + tan² x))/( tan x) = (-1 ± sec x))/( tan x)

Alternatively, we can use the sin and cos half-angle formulas, but then we must find sin and cos from tan.

Then, if we are in quadrant 3, tan of the half angle will be negative and tan x will be positive and sec x will be negative , so we use

(-1 + sec x))/( tan x)

As cot u = 1.878, tan u = 1/1.878 and sec u = -√(1 + (1/1.878)²)

tan x/2 = (-1 - √(1 + (1/1.878)²)/(1/1.878) =

1.878 (-1 - √(1 + (1/1.878)²) =

-1.878 - √(1 + 1.878²) =

-4.006