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I want to prove the asymptotes for the inverse cotangent.

  • #1
808
37

Homework Statement


arccot x = (π/2) - arctan x
arccot x =/= π
arccot x =/= 0

Homework Equations


arccot x = 1/arctan x (if x > 0)
arccot x = 1/arctan x + π (if x < 0)
arccot x = π/2 (if x = 0)

The Attempt at a Solution


π and 0 are the horizontal asymptotes, the values for which y (sine) cannot be.

arccot x = (π/2) - arctan x
arccot x =/= π
π =/= (π/2) - arctan x
(π/2) =/= -arctan x
-(π/2) =/= arctan x

arccot x = (π/2) - arctan x
arccot x =/= 0
0 =/= (π/2) - arctan x
-(π/2) =/= -arctan x
(π/2) =/= arctan x

Therefore, arctan x cannot be equal to negative or positive π/2, because if it were, then adding it to the original equation would produce π or 0, respectively. This cannot work because these are the values that it cannot be. As I write this, I'm thinking that I'm only establishing the fact that 0 and π are the asymptotes. But how would I prove that they're the asymptotes? If it's beyond the scope of precalculus, please say so.
 
Last edited:

Answers and Replies

  • #2
33,270
4,975

Homework Statement


arccot x = (π/2) - arctan x
arccot x =/= π
arccot x =/= 0

Homework Equations


arccot x = 1/arctan x (if x > 0)
arccot x = 1/arctan x + π (if x < 0)
arccot x = π/2 (if x = 0)

The Attempt at a Solution


π and 0 are the horizontal asymptotes, the values for which y (sine) cannot be.

arccot x = (π/2) - arctan x
arccot x =/= π
π =/= (π/2) - arctan x
(π/2) =/= -arctan x
-(π/2) =/= arctan x

arccot x = (π/2) - arctan x
arccot x =/= 0
0 =/= (π/2) - arctan x
-(π/2) =/= -arctan x
(π/2) =/= arctan x

Therefore, arctan x cannot be equal to negative or positive π/2, because if it were, then adding it to the original equation would produce π or 0, respectively. This cannot work because these are the values that it cannot be. As I write this, I'm thinking that I'm only establishing the fact that 0 and π are the asymptotes. But how would I prove that they're the asymptotes? If it's beyond the scope of precalculus, please say so.
What is there to prove? You can reflect the graph of y = cot(x) across the line y = x to get the graph of y = arccot(x).
 
Last edited:

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