The addition formulas for sine and cosine

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
• The normal component of the curl is the limiting circulation per unit area of shrinking discs 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
• Schwarz lanterns can have mesh tending to zero while their polyhedral areas diverge Counterexample
• Canonical Banach complexification of a real Banach space Lemma
• Finite sums of the sine harmonics Lemma
• Nearest-integer probe points for the Weierstrass function Lemma
• The finite Viete cosine product and its positive nested-radical factors Lemma
• The quaternion double cover generates the third homotopy group of SO(3) Lemma
• A disc of radius r has Riemann area pi r squared; in particular the unit disc has area pi Theorem
• 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
• Inscribed regular-polygon perimeters increase to 2 pi, while circumscribed perimeters decrease to 2 pi Theorem
• Product-to-sum and sum-to-product identities Theorem
• Quarter-turn values and shifts by pi/2 and pi Theorem
• The Gregory-Leibniz series: pi over four equals 1-1/3+1/5-1/7+... 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 · two levels

15 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)