Answer to Question #190729 in Algebra for OhMoe

Question

If α is in quadrant 4, which quadrant is arctan(cosα) in? 

Answer 1

α ∈ [ 3π/2 , 2π ]

So, cos α ≥ 0

i.e., cos α ∈ [0 ,1] ∀ α ∈ [ 3π/2 , 2π ]

On taking arctan of both sides we have

arctan ( cos α ) ∈ [0 , π /4 ] ∀ α ∈ [ 3π/2 , 2π ] [ ∵ arctan(1) = π /4

arctan(0) = 0 ]

We see that [ 0 , π /4 ] belongs to 1^st quadrant. Hence, if α is in quadrant 4, then arctan (cos α) always lies in 1^st quadrant.