Question

Find the principal solution of the following equation: 

tan θ = – 1

Answer 1

We know that,

`tan  pi/(4) = 1 and tan (pi - θ)` = – tan θ,
tan (2π – θ) = – tanθ

∴ `tan(pi - pi/4) = - tan  pi/(4)` = - 1

and `tan(2pi - pi/4) = -tan  pi/(4)` = – 1

∴ `tan  (3pi)/(4) = tan  (7pi)/(4)` = – 1, where

`0 < (3pi)/(4) < 2pi and 0 < (7pi)/(4) < 2pi`

∴ tan θ = – 1 gives,

tan θ = `tan  (3pi)/(4) = tan  (7pi)/(4)`

∴ θ = `(3pi)/(4)` and θ = `(7pi)/(4)`
Hence, the required principal solutions are

θ = `(3pi)/(4)` and θ = `(7pi)/(4)`.