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Signs, monotonicity intervals, and ranges of sine and cosine
Statement
Sine is strictly increasing on each interval and strictly decreasing on each interval . Cosine is strictly decreasing on and strictly increasing on . Both functions have range .
Facts & Assumptions
Given: An integer .
The zero sets, signs on the fundamental intervals, and period follow from The zero sets of sine and cosine and the least positive common period 2 pi.
Quarter-turn values give the endpoint values (Quarter-turn values and shifts by pi/2 and pi).
, , and the mean value theorem converts derivative sign into strict monotonicity (The derivatives of sine and cosine are cosine and minus sine, The mean value theorem, as the case of Cauchy's: for continuous on with and differentiable on there is with ).
Proof
On the open intervals where is positive respectively negative, makes sine strictly increasing respectively decreasing.
On the open intervals where is positive respectively negative, makes cosine strictly decreasing respectively increasing.
The period moves these conclusions to every integer , and the endpoint values in [L2] show both ranges are exactly .
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
- Principal inverse sine and inverse cosine Definition
- 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
- 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
- The hyperspherical-coordinate Jacobian is the standard product of a radial power and sine powers Example
- Tangent is a continuous strictly increasing bijection from (-π/2,π/2) onto ℝ Lemma
- 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
- t↦(cos t,sin t) is a bijection from [0,2π) onto the real unit circle Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 73 results over 24 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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)