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The addition formulas for sine and cosine
Statement
For all real ,
Facts & Assumptions
Given: A fixed real and variable real .
The harmonic-oscillator initial-value problem has the unique solution stated in Existence and uniqueness for y''=-y with prescribed initial data.
Sine and cosine have the derivatives and values at zero of The derivatives of sine and cosine are cosine and minus sine, and the chain rule applies (The chain rule, in one line from Carathéodory: if is differentiable at and is differentiable at , then is differentiable at with ).
Proof
The function solves with initial values , .
The function has the same equation and initial values.
Uniqueness in [L1] gives the sine addition formula.
Repeating the argument for and gives the cosine addition formula.
Depends on
- Existence and uniqueness for y''=-y with prescribed initial data
- The derivatives of sine and cosine are cosine and minus sine
- The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g(c)$, then $f \circ g$ is differentiable at $c$ with $(f \circ g)'(c) = f'(g(c))\,g'(c)$
Used by
- Every a cos x+b sin x has the amplitude-phase form R cos(x-φ) Corollary
- Parity and the Pythagorean identity for sine and cosine Corollary
- f(x+iy)=eˣ(cos 2y+i sin 2y) is continuous, satisfies f(z+w)=f(z)f(w) and f(1)=e, but is not the standard complex exponential Counterexample
- 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
- Double-angle and quadratic power-reduction identities Theorem
- Finite binomial formulas for cos(nθ) and sin(nθ) Theorem
- Product-to-sum and sum-to-product identities Theorem
- Quarter-turn values and shifts by pi/2 and pi Theorem
- The subtraction formulas for sine and cosine Theorem
- Tₙ(cosθ)=cos(nθ) and Uₙ(cosθ)sinθ=sin((n+1)θ) for every n∈ℕ Theorem
- Triple-angle identities for sine, cosine, and tangent Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 42 results over 19 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)