Double angle formulas

From Maths
Revision as of 18:42, 9 June 2017 by Alec (Talk | contribs) (Created page with "==Statement== The "''double angle formulas''" refer to the following two formulas * {{M|\forall \varphi,\psi\in\mathbb{R}[\sin(\varphi\pm\psi)\eq\sin(\varphi)\cos(\psi)\pm\cos...")

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

Statement

The "double angle formulas" refer to the following two formulas

  • [ilmath]\forall \varphi,\psi\in\mathbb{R}[\sin(\varphi\pm\psi)\eq\sin(\varphi)\cos(\psi)\pm\cos(\varphi)\sin(\psi)][/ilmath]
  • [ilmath]\forall\varphi,\psi\in\mathbb{R}[\cos(\varphi\pm\psi)\eq\cos(\varphi)\cos(\psi)\mp\sin(\varphi)\sin(\psi)][/ilmath]

However sometimes it is taken to mean the following two special cases:

  • [ilmath]\forall\varphi\in\mathbb{R}[\sin(2\varphi)\eq2\sin(\varphi)\cos(\varphi)][/ilmath] and
  • [ilmath]\forall\varphi\in\mathbb{R}\big[\cos(2\varphi)\eq(\cos(\varphi))^2-(\sin(\varphi))^2\big][/ilmath]
    • Noting that [ilmath](\sin(\theta))^2+(\cos(\theta))^2\eq 1[/ilmath] we see that [ilmath](\cos(\theta))^2\eq 1-(\sin(\theta))^2[/ilmath] and [ilmath](\sin(\theta))^2\eq 1-(\cos(\theta))^2[/ilmath], yielding:
      1. [ilmath]\forall\varphi\in\mathbb{R}\big[\cos(2\varphi)\eq 1-2(\sin(\varphi))^2\big][/ilmath] and
      2. [ilmath]\forall\varphi\in\mathbb{R}\big[\cos(2\varphi)\eq 2(\cos(\varphi))^2-1\big][/ilmath]
    Either form is commonplace.