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The derivatives of sine and cosine are cosine and minus sine
Statement
The functions and are differentiable on , with Also and .
Facts & Assumptions
Given: The sine and cosine power series.
Both defining series converge on all of (The sine and cosine power series converge absolutely for every real argument, Sine and cosine defined by their real power series).
A real power series may be differentiated term by term inside its radius of convergence (Inside its radius a real power series may be differentiated term by term, and the differentiated series has the same radius).
Proof
Termwise differentiation of the sine series gives .
Termwise differentiation of the cosine series gives .
Evaluating the defining series at gives and .
Depends on
Used by
- Parity and the Pythagorean identity for sine and cosine Corollary
- The limit of sin x divided by x at zero is one Corollary
- An invertible derivative at one point does not give a local inverse without C¹ regularity Counterexample
- f(x+iy)=eˣ(cos 2y+isin 2y) is continuous, satisfies f(z+w)=f(z)f(w) and f(1)=e, but is not the standard complex exponential Counterexample
- Principal arcsine has no finite derivative at -1 or 1 Counterexample
- Principal inverse sine and inverse cosine Definition
- Cylindrical coordinates have absolute Jacobian determinant r on an injective compact box Example
- Machin's formula π/4=4arctan(1/5)-arctan(1/239) 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 r²sinφ away from the axis and angular seam Example
- The extension of x² sin(1/x) by zero is differentiable but its derivative is discontinuous at zero Example
- The hyperspherical-coordinate Jacobian is the standard product of a radial power and sine powers Example
- Sine is positive and cosine is strictly decreasing on (0,2), with cos 2 at most -1/3 Lemma
- Derivatives and fundamental periods of tangent, cotangent, secant, and cosecant Theorem
- Existence and uniqueness for y''=-y with prescribed initial data Theorem
- For -1<y<1, (arcsin y)ᵖʳⁱᵐᵉ=1/√1-y² and (arccos y)ᵖʳⁱᵐᵉ=-1/√1-y² Theorem
- Principal arctangent: derivative, integral, power series, and the Gregory–Leibniz series Theorem
- Signs, monotonicity intervals, and ranges of sine and cosine Theorem
- The addition formulas for sine and cosine Theorem
- There is no continuous logarithm on all of ℂ∖{0} Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 58 results over 11 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)