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Derivatives and fundamental periods of tangent, cotangent, secant, and cosecant
Statement
On their natural domains, Tangent and cotangent have least positive period ; secant and cosecant have least positive period .
Facts & Assumptions
Given: A real in the domain of the function under discussion.
The quotient definitions and exact excluded points are those of Tangent, cotangent, secant, and cosecant on their exact natural domains.
Quotient and reciprocal differentiation rules apply, and the zero sets and periods of sine/cosine are known (Sums, scalar multiples, products and quotients: , , , and when , The zero sets of sine and cosine and the least positive common period 2 pi).
Proof
Quotient and reciprocal differentiation applied to [L1] gives the four displayed derivatives after using .
Shifting by negates both sine and cosine, so their quotients have period , while their reciprocals change sign and therefore have period .
The zero-set and quarter-turn values rule out a smaller positive period in each case.
Depends on
- Tangent, cotangent, secant, and cosecant on their exact natural domains
- The derivatives of sine and cosine are cosine and minus sine
- Parity and the Pythagorean identity for sine and cosine
- Sums, scalar multiples, products and quotients: $(f+g)'(c) = f'(c) + g'(c)$, $(\alpha f)'(c) = \alpha f'(c)$, $(fg)'(c) = f'(c)g(c) + f(c)g'(c)$, and $(f/g)'(c) = \bigl(f'(c)g(c) - f(c)g'(c)\bigr)/g(c)^{2}$ when $g(c) \ne 0$
- The zero sets of sine and cosine and the least positive common period 2 pi
Used by
- Tangent is a continuous strictly increasing bijection from (-π/2,π/2) onto ℝ Lemma
- 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
- The Gregory-Leibniz series: pi over four equals 1-1/3+1/5-1/7+... Theorem
Dependency tree · two levels
25 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)