Return to navigation page or list derivations

Review identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation

step inference rule input feed output step validity (as per SymPy)
1
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 2103023049
    \(\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)\)
 no validation is available for declarations
2
  • 0000111886: change variable X to Y
  • number of inputs: 1; feeds: 2; outputs: 1
  • Change variable $#1$ to $#2$ in Eq.~\ref{eq:#3}; yields Eq.~\ref{eq:#4}.
  1. 2103023049
    \(\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)\)
  1. 9110536742
    \(2 x\)
  2. 4961662865
    \(x\)
  1. 8483686863
    \(\sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x) \right)\)
 LHS diff is sin(pdg0001464) - sin(2*pdg0001464) RHS diff is (-exp(4*pdg0001464*pdg0004621) + exp(3*pdg0001464*pdg0004621) - exp(pdg0001464*pdg0004621) + 1)*exp(-2*pdg0001464*pdg0004621)/(2*pdg0004621)
3
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 4585932229
    \(\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)\)
 no validation is available for declarations
4
  • 0000111253: multiply expr 1 by expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Multiply Eq.~\ref{eq:#1} by Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 4585932229
    \(\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)\)
  2. 2103023049
    \(\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)\)
  1. 3470587782
    \(\sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)\)
 valid
5
  • 0000111182: multiply both sides by
  • number of inputs: 1; feeds: 1; outputs: 1
  • Multiply both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.
  1. 3470587782
    \(\sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)\)
  1. 8642992037
    \(2\)
  1. 9894826550
    \(2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \left(\exp(i x)+\exp(-i x) \right)\)
 valid
6
  • 0000111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 9894826550
    \(2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \left(\exp(i x)+\exp(-i x) \right)\)
  1. 8699789241
    \(2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)\)
 valid
7
  • 0000111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 8699789241
    \(2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)\)
  1. 9180861128
    \(2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right)\)
 valid
8
  • 0000111863: RHS of expr 1 equals RHS of expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • RHS of Eq.~\ref{eq:#1} is equal to RHS of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 8483686863
    \(\sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x) \right)\)
  2. 9180861128
    \(2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right)\)
  1. 2405307372
    \(\sin(2 x) = 2 \sin(x) \cos(x)\)
 valid
9
  • 0000111341: declare final expression
  • number of inputs: 1; feeds: 0; outputs: 0
  • Eq.~\ref{eq:#1} is one of the final equations.
  1. 2405307372
    \(\sin(2 x) = 2 \sin(x) \cos(x)\)
no validation is available for declarations

Symbols used in identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation

Steps and expressions for identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation

d3js visualization of steps and expressions in identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation


Hold the mouse over a node to highlight that node and its neighbors. You can zoom in/out. You can pan the image. You can move nodes by clicking and dragging.

Actions: Edit Derivation

You must be logged in to edit the steps of derivation, edit the abstract, or rename the derivation.

Generate Tex file or PDF file
   xor   

Delete Derivation and all associated steps

You must be logged in to delete the derivation.

timing of Neo4j queries: