Alphabeta MathHow statement and proof provenance work
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Tangent, cotangent, secant, and cosecant on their exact natural domains
Definition
Define where ranges over the integers. The exclusions are exactly the zero sets of The zero sets of sine and cosine and the least positive common period 2 pi.
Depends on
Used by
- u=v=π/4 shows that the tangent addition formula cannot omit its domain restrictions Counterexample
- Addition and half-angle identities compute the sine, cosine, and tangent of π/12 Example
- Machin's formula π/4=4 arctan(1/5)-arctan(1/239) Example
- Tangent is a continuous strictly increasing bijection from (-π/2,π/2) onto ℝ Lemma
- Polar-coordinate forms of two Cartesian expressions Proposition
- Addition and subtraction formulas for tangent, cotangent, secant, and cosecant on their exact domains Theorem
- Cofunction, supplementary, quarter-turn, and reflection identities for the six trigonometric functions Theorem
- Derivatives and fundamental periods of tangent, cotangent, secant, and cosecant Theorem
- Double-angle and quadratic power-reduction identities Theorem
- Half-angle identities with the sign determined by the quadrant Theorem
- Inscribed regular-polygon perimeters increase to 2 pi, while circumscribed perimeters decrease to 2 pi Theorem
- Principal arctangent: derivative, integral, power series, and the Gregory–Leibniz series Theorem
- Pythagorean and parity identities for all six trigonometric functions on their natural domains Theorem
- The Gregory-Leibniz series: pi over four equals 1-1/3+1/5-1/7+... Theorem
- The Mittag-Leffler expansion of pi cotangent Theorem
- The tangent half-angle identities and rational parametrization of the unit circle away from (-1,0) Theorem
- Triple-angle identities for sine, cosine, and tangent Theorem
Dependency tree · two levels
7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- NIST Digital Library of Mathematical Functions, Chapter 4 (standard reference, not scraped)
- C. Schmeiser, Introduction to Analysis (standard reference, not scraped)