Physics Derivation Graph: review derivation: Euler equation: trigonometric relations

review derivation: Euler equation: trigonometric relations

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Physics Derivation Graph: Steps for Euler equation: trigonometric relations
Index	Inference Rule	Input latex	Feeds latex	Output latex	step validity	dimension check	unit check
6	divide both sides by	4742644828; locally 2939484: $\exp (i x) + \exp (- i x) = 2 \cos (x)$ $e^{p d g_{1464} p d g_{4621}} + e^{- p d g_{1464} p d g_{4621}} = 2 \cos (p d g_{1464})$	0004829194: $2$ $2$	3829492824; locally 4383592: $\frac{1}{2} (\exp (i x) + \exp (- i x)) = \cos (x)$ $\frac{e^{p d g_{1464} p d g_{4621}}}{2} + \frac{e^{- p d g_{1464} p d g_{4621}}}{2} = \cos (p d g_{1464})$	valid	4742644828: 3829492824:	4742644828: 3829492824:
8	declare final expr	4585932229; locally 4849888: $\cos (x) = \frac{1}{2} (\exp (i x) + \exp (- i x))$ $\cos (p d g_{1464}) = \frac{e^{p d g_{1464} p d g_{4621}}}{2} + \frac{e^{- p d g_{1464} p d g_{4621}}}{2}$ 2103023049; locally 4849959: $\sin (x) = \frac{1}{2 i} (\exp (i x) - \exp (- i x))$ $\sin (p d g_{1464}) = \frac{e^{p d g_{1464} p d g_{4621}} - e^{- p d g_{1464} p d g_{4621}}}{2 p d g_{4621}}$			no validation is available for declarations	4585932229: 2103023049:	4585932229: 2103023049:
11	divide both sides by	3942849294; locally 4825483: $\exp (i x) - \exp (- i x) = 2 i \sin (x)$ $e^{p d g_{1464} p d g_{4621}} - e^{- p d g_{1464} p d g_{4621}} = 2 p d g_{4621} \sin (p d g_{1464})$	0001921933: $2 i$ $2 p d g_{4621}$	4843995999; locally 1133483: $\frac{1}{2 i} (\exp (i x) - \exp (- i x)) = \sin (x)$ $\frac{e^{p d g_{1464} p d g_{4621}} - e^{- p d g_{1464} p d g_{4621}}}{2 p d g_{4621}} = \sin (p d g_{1464})$	valid	3942849294: 4843995999:	3942849294: 4843995999:
10	add expr 1 to expr 2	4938429483; locally 8888888: $\exp (i x) = \cos (x) + i \sin (x)$ $e^{p d g_{1464} p d g_{4621}} = p d g_{4621} \sin (p d g_{1464}) + \cos (p d g_{1464})$ 2123139121; locally 3194924: $- \exp (- i x) = - \cos (x) + i \sin (x)$ $- e^{- p d g_{1464} p d g_{4621}} = p d g_{4621} \sin (p d g_{1464}) - \cos (p d g_{1464})$		3942849294; locally 4825483: $\exp (i x) - \exp (- i x) = 2 i \sin (x)$ $e^{p d g_{1464} p d g_{4621}} - e^{- p d g_{1464} p d g_{4621}} = 2 p d g_{4621} \sin (p d g_{1464})$	valid	4938429483: 2123139121: 3942849294:	4938429483: 2123139121: 3942849294:
7	swap LHS with RHS	3829492824; locally 4383592: $\frac{1}{2} (\exp (i x) + \exp (- i x)) = \cos (x)$ $\frac{e^{p d g_{1464} p d g_{4621}}}{2} + \frac{e^{- p d g_{1464} p d g_{4621}}}{2} = \cos (p d g_{1464})$		4585932229; locally 4849888: $\cos (x) = \frac{1}{2} (\exp (i x) + \exp (- i x))$ $\cos (p d g_{1464}) = \frac{e^{p d g_{1464} p d g_{4621}}}{2} + \frac{e^{- p d g_{1464} p d g_{4621}}}{2}$	valid	3829492824: 4585932229:	3829492824: 4585932229:
5	add expr 1 to expr 2	4938429483; locally 8888888: $\exp (i x) = \cos (x) + i \sin (x)$ $e^{p d g_{1464} p d g_{4621}} = p d g_{4621} \sin (p d g_{1464}) + \cos (p d g_{1464})$ 4938429484; locally 8888883: $\exp (- i x) = \cos (x) - i \sin (x)$ $e^{- p d g_{1464} p d g_{4621}} = - p d g_{4621} \sin (p d g_{1464}) + \cos (p d g_{1464})$		4742644828; locally 2939484: $\exp (i x) + \exp (- i x) = 2 \cos (x)$ $e^{p d g_{1464} p d g_{4621}} + e^{- p d g_{1464} p d g_{4621}} = 2 \cos (p d g_{1464})$	valid	4938429483: 4938429484: 4742644828:	4938429483: 4938429484: 4742644828:
9	multiply both sides by	4938429484; locally 8888883: $\exp (- i x) = \cos (x) - i \sin (x)$ $e^{- p d g_{1464} p d g_{4621}} = - p d g_{4621} \sin (p d g_{1464}) + \cos (p d g_{1464})$	0003747849: $- 1$ $- 1$	2123139121; locally 3194924: $- \exp (- i x) = - \cos (x) + i \sin (x)$ $- e^{- p d g_{1464} p d g_{4621}} = p d g_{4621} \sin (p d g_{1464}) - \cos (p d g_{1464})$	valid	4938429484: 2123139121:	4938429484: 2123139121:
2	change variable X to Y	4938429483; locally 8888888: $\exp (i x) = \cos (x) + i \sin (x)$ $e^{p d g_{1464} p d g_{4621}} = p d g_{4621} \sin (p d g_{1464}) + \cos (p d g_{1464})$	0002393922: $x$ $p d g_{1464}$ 0003949052: $- x$ $- p d g_{1464}$	2394853829; locally 8888881: $\exp (- i x) = \cos (- x) + i \sin (- x)$ $e^{- p d g_{1464} p d g_{4621}} = - p d g_{4621} \sin (p d g_{1464}) + \cos (p d g_{1464})$	valid	4938429483: 2394853829: error for dim with 2394853829	4938429483: 2394853829: N/A
12	swap LHS with RHS	4843995999; locally 1133483: $\frac{1}{2 i} (\exp (i x) - \exp (- i x)) = \sin (x)$ $\frac{e^{p d g_{1464} p d g_{4621}} - e^{- p d g_{1464} p d g_{4621}}}{2 p d g_{4621}} = \sin (p d g_{1464})$		2103023049; locally 4849959: $\sin (x) = \frac{1}{2 i} (\exp (i x) - \exp (- i x))$ $\sin (p d g_{1464}) = \frac{e^{p d g_{1464} p d g_{4621}} - e^{- p d g_{1464} p d g_{4621}}}{2 p d g_{4621}}$	valid	4843995999: 2103023049:	4843995999: 2103023049:
4	function is odd	4938429482; locally 8888882: $\exp (- i x) = \cos (x) + i \sin (- x)$ $e^{- p d g_{1464} p d g_{4621}} = - p d g_{4621} \sin (p d g_{1464}) + \cos (p d g_{1464})$	0003919391: $x$ $p d g_{1464}$ 0003981813: $- \sin (x)$ $- \sin (p d g_{1464})$ 0002919191: $\sin (- x)$ $- \sin (p d g_{1464})$	4938429484; locally 8888883: $\exp (- i x) = \cos (x) - i \sin (x)$ $e^{- p d g_{1464} p d g_{4621}} = - p d g_{4621} \sin (p d g_{1464}) + \cos (p d g_{1464})$	no check performed	4938429482: error for dim with 4938429482 4938429484:	4938429482: N/A 4938429484:
1	declare initial expr			4938429483; locally 8888888: $\exp (i x) = \cos (x) + i \sin (x)$ $e^{p d g_{1464} p d g_{4621}} = p d g_{4621} \sin (p d g_{1464}) + \cos (p d g_{1464})$	no validation is available for declarations	4938429483:	4938429483:
3	function is even	2394853829; locally 8888881: $\exp (- i x) = \cos (- x) + i \sin (- x)$ $e^{- p d g_{1464} p d g_{4621}} = - p d g_{4621} \sin (p d g_{1464}) + \cos (p d g_{1464})$	0004849392: $x$ $p d g_{1464}$ 0001030901: $\cos (x)$ $\cos (p d g_{1464})$ 0003413423: $\cos (- x)$ $\cos (p d g_{1464})$	4938429482; locally 8888882: $\exp (- i x) = \cos (x) + i \sin (- x)$ $e^{- p d g_{1464} p d g_{4621}} = - p d g_{4621} \sin (p d g_{1464}) + \cos (p d g_{1464})$	no check performed	2394853829: error for dim with 2394853829 4938429482: error for dim with 4938429482	2394853829: N/A 4938429482: N/A

symbol ID	category	latex	scope	dimension	name	value	Used in derivations	references
1464	variable	x $x$	['real']	dimensionless			double intensity when phase is coherent (optics) time invariant force conserves energy Euler equation proof angle of maximum distance for projectile motion quadratic equation derivation identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation optics: Law of refraction to Brewster's angle quantum basics orthogonality Euler equation to e^(i pi) + 1 = 0 particle in a 1D box Euler equation: trig square root hyperbolic trigonometric identities variance relation Euler equation: trigonometric relations equations of motion in 2D (calculus) Lorentz transformation		140
4621	variable	i $i$	['imaginary']	dimensionless	imaginary unit		double intensity when phase is coherent (optics) Euler equation proof identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation derivation of Schrodinger Equation electric field wave equation: from time dependent to time independent Euler equation to e^(i pi) + 1 = 0 Euler equation: trig square root hyperbolic trigonometric identities Euler equation: trigonometric relations	Imaginary_unit	74

review derivation: Euler equation: trigonometric relations

Symbols for this derivation