Math 2.0

Special functions Definition

Arctangent

Builds on Tangent functionContinuity of the inverse of a strictly monotone continuous function

Definition

The arctangent is the function arctan⁡ ⁣:ℝ→(−π/2,π/2)\arctan \colon \mathbb{R} \to (-\pi/2, \pi/2) such that, for each real xx, arctan⁡x\arctan x is the unique y∈(−π/2,π/2)y \in (-\pi/2, \pi/2) with tan⁡y=x\tan y = x.

Remarks

The tangent is strictly increasing and continuous on (−π/2,π/2)(-\pi/2, \pi/2), with range ℝ\mathbb{R}. So arctan⁡\arctan is well defined and continuous. For example, arctan⁡0=0\arctan 0 = 0 and arctan⁡1=π/4\arctan 1 = \pi/4. Also arctan⁡x→π/2\arctan x \to \pi/2 as x→∞x \to \infty, and arctan⁡x→−π/2\arctan x \to -\pi/2 as x→−∞x \to -\infty. Mathlib defines it as Real.arctan.

Reading path

To understand this page from the ground up, read these 20 pages in order. The foundations come first.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionSequence, definition, no evidence levelSequencedefinitionStrictly monotone function, definition, no evidence levelStrictly monotone fun…definitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionReal sequence, definition, no evidence levelReal sequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionOne-sided limit, definition, no evidence levelOne-sided limitdefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementPower series, definition, no evidence levelPower seriesdefinitionIntermediate value theorem, statement, no evidence levelIntermediate value th…statementOne-sided limits of a monotone function exist, statement, no evidence levelOne-sided limits of a…statementSine and cosine, definition, no evidence levelSine and cosinedefinitionContinuity of the inverse of a strictly monotone continuous function, statement, no evidence levelContinuity of the inv…statementTangent function, definition, no evidence levelTangent functiondefinitionArctangent, definition, no evidence levelArctangentdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Limit point of a set definition
  2. Sequence definition
  3. Strictly monotone function definition
  4. Limit of a function definition
  5. Real sequence definition
  6. Continuity at a point definition
  7. One-sided limit definition
  8. Bounded sequence definition
  9. Limit of a sequence definition
  10. Monotone sequence definition
  11. Series definition
  12. Continuous function definition
  13. Monotone convergence theorem for sequences statement
  14. Power series definition
  15. Intermediate value theorem statement
  16. One-sided limits of a monotone function exist statement
  17. Sine and cosine definition
  18. Continuity of the inverse of a strictly monotone continuous function statement
  19. Tangent function definition
  20. Arctangent definition

Where this leads

4 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Inverse_trigonometric_functions, Principal values
About this page
Tag065XW2V6C3
URIm2://065XW2V6C3
Kinddefinition
WritingW0 stub
MSC33B10
DefinesArctangent
UsesTangent function, Continuity of the inverse of a strictly monotone continuous function

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