Signs, monotonicity intervals, and ranges of sine and cosine

Depends on

• The zero sets of sine and cosine and the least positive common period 2 pi
• Quarter-turn values and shifts by pi/2 and pi
• The derivatives of sine and cosine are cosine and minus sine
• The mean value theorem, as the case $g(x) = x$ of Cauchy's: for $f$ continuous on $[a,b]$ with $a < b$ and differentiable on $(a,b)$ there is $c \in (a,b)$ with $f(b) - f(a) = f'(c)(b-a)$

Used by

• An invertible derivative at one point does not give a local inverse without C¹ regularity Counterexample
• Antipodal points on a round sphere have many minimizing geodesics Counterexample
• Principal inverse sine and inverse cosine Definition
• A right circular cylinder is an elementary solid region, presented by two caps and four side quarters Example
• Cylindrical coordinates have absolute Jacobian determinant r on an injective compact box Example
• Exact sine and cosine values at π/10, π/5, and 2π/5 Example
• Normal coordinates on the round sphere Example
• Polar change of variables on a compact annular sector gives the Jacobian factor r and its area Example
• Spherical coordinates have absolute Jacobian determinant r² sinφ away from the axis and angular seam Example
• Stokes' theorem on a flat disc and on a hemisphere with the same induced boundary circle Example
• Surface area and flux on a sphere, with scalar integrals on a hemisphere Example
• The closed ball is an elementary solid region, presented by the eight spherical octants Example
• The hyperspherical-coordinate Jacobian is the standard product of a radial power and sine powers Example
• The Mobius band presented by two regular patches, with normal comparison on the interiors of the overlap components Example
• The outward flux of the inverse-square field through a sphere centred at the origin is 4π Example
• The surface area of a torus is 4π²ab Example
• FALSE: an invertible derivative at one point gives a local inverse False statement
• Normal coordinates make the metric Euclidean throughout the chart False statement
• Tangent is a continuous strictly increasing bijection from (-π/2,π/2) onto ℝ Lemma
• The finite Viete cosine product and its positive nested-radical factors Lemma
• The topologist's sine curve is connected Lemma
• The Weierstrass tail has one sign and dominates at the probe points Lemma
• Wallis integrals satisfy the two-step recurrence, closed forms, and the adjacent-integral squeeze Lemma
• Symmetry and the trigonometric form of the real Beta integral Proposition
• For -1<y<1, (arcsin y)ᵖʳⁱᵐᵉ=1/√1-y² and (arccos y)ᵖʳⁱᵐᵉ=-1/√1-y² Theorem
• For n≥1, 2¹⁻ⁿTₙ is the minimax monic polynomial of degree n on [-1,1] 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
• t↦(cos t,sin t) is a bijection from [0,2π) onto the real unit circle Theorem
• The Gregory-Leibniz series: pi over four equals 1-1/3+1/5-1/7+... Theorem

Dependency tree · two levels

23 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)