![]() | 1 | initial version |
You got:
sage: var("theta phi M_1 M_2")
sage: %display typeset
sage: f = (sin(theta+phi)-M_1/M_2*sin(phi)).simplify_full()
sage: f, solve(f, phi)
Which is: $$ \newcommand{\Bold}[1]{\mathbf{#1}}\left(\frac{{\left(\cos\left(\theta\right) \sin\left(\phi\right) + \cos\left(\phi\right) \sin\left(\theta\right)\right)} M_{2} - M_{1} \sin\left(\phi\right)}{M_{2}}, \left[\sin\left(\phi\right) = -\frac{M_{2} \cos\left(\phi\right) \sin\left(\theta\right)}{M_{2} \cos\left(\theta\right) - M_{1}}\right]\right) $$
Hence, the solution for $\phi$ is obtianed by intersecting the tangent function with a constant line (in the interval $[\pi, \pi]$): - if $\theta$ is an integer multiple of $\pi\2$, then $\phi = {0,\pm \pi}$ is the solution set. - in general, there are exactly $2$ intersections. - if the values of $M_1$ and $M_2$ allow it ($|M_1/M_2| \leq 1$), then we have no solution.
Remark. I'd be interested to know if there is a way to do this sort of "case study" automatically. I tried with some solver options (explicit, etc.), but it doesn't help. Neither with sympy solve or solveset.. but maybe I'm missing to pass some extra assumptions.
![]() | 2 | No.2 Revision |
You got:
sage: var("theta phi M_1 M_2")
sage: %display typeset
sage: f = (sin(theta+phi)-M_1/M_2*sin(phi)).simplify_full()
sage: f, solve(f, phi)
Which is: $$ \newcommand{\Bold}[1]{\mathbf{#1}}\left(\frac{{\left(\cos\left(\theta\right) \sin\left(\phi\right) + \cos\left(\phi\right) \sin\left(\theta\right)\right)} M_{2} - M_{1} \sin\left(\phi\right)}{M_{2}}, \left[\sin\left(\phi\right) = -\frac{M_{2} \cos\left(\phi\right) \sin\left(\theta\right)}{M_{2} \cos\left(\theta\right) - M_{1}}\right]\right) $$
Hence, the solution for $\phi$ is obtianed by intersecting the tangent function with a constant line (in the interval $[\pi, \pi]$):
- if $\theta$ is an integer multiple of $\pi\2$, $\pi/2$, then $\phi = {0,\pm \pi}$ is the solution set.
- in general, there are exactly $2$ intersections.
- if the values of $M_1$ and $M_2$ allow it ($|M_1/M_2| \leq 1$), then we have no solution.
Remark. I'd be interested to know if there is a way to do this sort of "case study" automatically. I tried with some solver options (explicit, etc.), but it doesn't help. Neither with sympy solve or solveset.. but maybe I'm missing to pass some extra assumptions. assumptions.
![]() | 3 | No.3 Revision |
You got:
sage: var("theta phi M_1 M_2")
sage: %display typeset
sage: f = (sin(theta+phi)-M_1/M_2*sin(phi)).simplify_full()
sage: f, solve(f, phi)
Which is: $$ \newcommand{\Bold}[1]{\mathbf{#1}}\left(\frac{{\left(\cos\left(\theta\right) \sin\left(\phi\right) + \cos\left(\phi\right) \sin\left(\theta\right)\right)} M_{2} - M_{1} \sin\left(\phi\right)}{M_{2}}, \left[\sin\left(\phi\right) = -\frac{M_{2} \cos\left(\phi\right) \sin\left(\theta\right)}{M_{2} \cos\left(\theta\right) - M_{1}}\right]\right) $$
Hence, the solution for $\phi$ is obtianed by intersecting the tangent function with a constant line (in line, and by periodicity we can reason in the interval $[\pi, \pi]$):
- \pi]$:
Remark. I'd be interested to know if there is a way to do this sort of "case study" automatically. I tried with some solver options (explicit, etc.), but it doesn't help. Neither with sympy solve or solveset.. but maybe I'm missing to pass some extra assumptions.
![]() | 4 | No.4 Revision |
You got:
sage: var("theta phi M_1 M_2")
sage: %display typeset
sage: f = (sin(theta+phi)-M_1/M_2*sin(phi)).simplify_full()
sage: f, solve(f, phi)
Which is: $$ \newcommand{\Bold}[1]{\mathbf{#1}}\left(\frac{{\left(\cos\left(\theta\right) \sin\left(\phi\right) + \cos\left(\phi\right) \sin\left(\theta\right)\right)} M_{2} - M_{1} \sin\left(\phi\right)}{M_{2}}, \left[\sin\left(\phi\right) = -\frac{M_{2} \cos\left(\phi\right) \sin\left(\theta\right)}{M_{2} \cos\left(\theta\right) - M_{1}}\right]\right) $$
Hence, the solution for $\phi$ is obtianed by intersecting the tangent function with a constant line, and by periodicity we can reason in the interval $[\pi, $[-\pi, \pi]$:
Remark. I'd be interested to know if there is a way to do this sort of "case study" automatically. I tried with some solver options (explicit, etc.), but it doesn't help. Neither with sympy solve or solveset.. but maybe I'm missing to pass some extra assumptions.
![]() | 5 | No.5 Revision |
You got:
sage: var("theta phi M_1 M_2")
sage: %display typeset
sage: f = (sin(theta+phi)-M_1/M_2*sin(phi)).simplify_full()
sage: f, solve(f, phi)
Which is: $$ \newcommand{\Bold}[1]{\mathbf{#1}}\left(\frac{{\left(\cos\left(\theta\right) \sin\left(\phi\right) + \cos\left(\phi\right) \sin\left(\theta\right)\right)} M_{2} - M_{1} \sin\left(\phi\right)}{M_{2}}, \left[\sin\left(\phi\right) = -\frac{M_{2} \cos\left(\phi\right) \sin\left(\theta\right)}{M_{2} \cos\left(\theta\right) - M_{1}}\right]\right) $$
Hence, the solution for $\phi$ is obtianed by intersecting the tangent function with a constant line, and by periodicity we can reason in the interval $[-\pi, \pi]$:
Remark. I'd be interested to know if there is a way to do this sort of "case study" automatically. I tried with some solver options (explicit, etc.), but it doesn't help. Neither with sympy solve or solveset.. but maybe I'm missing to pass some extra assumptions.
![]() | 6 | No.6 Revision |
You got:
sage: var("theta phi M_1 M_2")
sage: %display typeset
sage: f = (sin(theta+phi)-M_1/M_2*sin(phi)).simplify_full()
sage: f, solve(f, phi)
Which is: $$ \newcommand{\Bold}[1]{\mathbf{#1}}\left(\frac{{\left(\cos\left(\theta\right) \sin\left(\phi\right) + \cos\left(\phi\right) \sin\left(\theta\right)\right)} M_{2} - M_{1} \sin\left(\phi\right)}{M_{2}}, \left[\sin\left(\phi\right) = -\frac{M_{2} \cos\left(\phi\right) \sin\left(\theta\right)}{M_{2} \cos\left(\theta\right) - M_{1}}\right]\right) $$
Hence, the solution for $\phi$ is obtianed by intersecting the tangent function with a constant line, and by periodicity we can reason in the interval $[-\pi, \pi]$:
Remark. I'd be interested to know if there is a way to do this sort of "case study" automatically. I tried with some solver options (explicit, etc.), but it doesn't help. Neither with sympy solve or solveset.. but maybe I'm missing to pass some extra assumptions.
![]() | 7 | No.7 Revision |
You got:
sage: var("theta phi M_1 M_2")
sage: %display typeset
sage: f = (sin(theta+phi)-M_1/M_2*sin(phi)).simplify_full()
sage: f, solve(f, phi)
Which is: $$ \newcommand{\Bold}[1]{\mathbf{#1}}\left(\frac{{\left(\cos\left(\theta\right) \sin\left(\phi\right) + \cos\left(\phi\right) \sin\left(\theta\right)\right)} M_{2} - M_{1} \sin\left(\phi\right)}{M_{2}}, \left[\sin\left(\phi\right) = -\frac{M_{2} \cos\left(\phi\right) \sin\left(\theta\right)}{M_{2} \cos\left(\theta\right) - M_{1}}\right]\right) $$
Hence, the solution for $\phi$ is obtianed by intersecting the tangent function with a constant line, and by periodicity we can reason in the interval $[-\pi, \pi]$:\pi]$, recall that:
Remark. I'd be interested to know if there is a way to do this sort of "case study" automatically. I tried with some solver options (explicit, etc.), but it doesn't help. Neither with sympy solve or solveset.. but maybe I'm missing to pass some extra assumptions.
![]() | 8 | No.8 Revision |
You got:
sage: var("theta phi M_1 M_2")
sage: %display typeset
sage: f = (sin(theta+phi)-M_1/M_2*sin(phi)).simplify_full()
sage: f, solve(f, phi)
Which is: $$ \newcommand{\Bold}[1]{\mathbf{#1}}\left(\frac{{\left(\cos\left(\theta\right) \sin\left(\phi\right) + \cos\left(\phi\right) \sin\left(\theta\right)\right)} M_{2} - M_{1} \sin\left(\phi\right)}{M_{2}}, \left[\sin\left(\phi\right) = -\frac{M_{2} \cos\left(\phi\right) \sin\left(\theta\right)}{M_{2} \cos\left(\theta\right) - M_{1}}\right]\right) $$
Hence, the solution for $\phi$ is obtianed by intersecting the tangent function with a constant line, and by periodicity we can reason in the interval $[-\pi, \pi]$, recall that:
Remark. I'd be interested to know if there is a way to do this sort of "case study" automatically. I tried with some solver options (explicit, etc.), but it doesn't help. Neither with sympy solve or solveset.. but maybe I'm missing to pass some extra assumptions.
![]() | 9 | No.9 Revision |
You got:
sage: var("theta phi M_1 M_2")
sage: f = (sin(theta+phi)-M_1/M_2*sin(phi)).simplify_full()
sage: f, solve(f, phi)
Which is: $$ \newcommand{\Bold}[1]{\mathbf{#1}}\left(\frac{{\left(\cos\left(\theta\right) \sin\left(\phi\right) + \cos\left(\phi\right) \sin\left(\theta\right)\right)} M_{2} - M_{1} \sin\left(\phi\right)}{M_{2}}, \left[\sin\left(\phi\right) = -\frac{M_{2} \cos\left(\phi\right) \sin\left(\theta\right)}{M_{2} \cos\left(\theta\right) - M_{1}}\right]\right) $$
Hence, the solution for $\phi$ is obtianed by intersecting the tangent function with a constant line, and by periodicity we can reason in the interval $[-\pi, \pi]$, recall that:
Remark. I'd be interested to know if there is a way to do this sort of "case study" automatically. I tried with some solver options (explicit, etc.), but it doesn't help. Neither with sympy SymPy solve or solveset.. the new solveset.. but maybe I'm missing to pass some extra assumptions.information.
![]() | 10 | No.10 Revision |
You got:
sage: var("theta phi M_1 M_2")
sage: f = (sin(theta+phi)-M_1/M_2*sin(phi)).simplify_full()
sage: f, solve(f, phi)
Which is: $$ \newcommand{\Bold}[1]{\mathbf{#1}}\left(\frac{{\left(\cos\left(\theta\right) \sin\left(\phi\right) + \cos\left(\phi\right) \sin\left(\theta\right)\right)} M_{2} - M_{1} \sin\left(\phi\right)}{M_{2}}, \left[\sin\left(\phi\right) = -\frac{M_{2} \cos\left(\phi\right) \sin\left(\theta\right)}{M_{2} \cos\left(\theta\right) - M_{1}}\right]\right) $$
Hence, the solution for $\phi$ is obtianed by intersecting the tangent function with a constant line, and by periodicity we can reason in the interval $[-\pi, \pi]$, recall that:
Remark. I'd be interested to know if there is a way to do this sort of "case study" automatically. I tried with some solver options (explicit, etc.), but it doesn't help. Neither with SymPy solve or the new solveset.. but maybe I'm missing to pass some extra information.
![]() | 11 | No.11 Revision |
You It seems to me that the hard work is already done properly by the solver. Finally, you got:
sage: var("theta phi M_1 M_2")
sage: f = (sin(theta+phi)-M_1/M_2*sin(phi)).simplify_full()
sage: f, solve(f, phi)
Which is: $$ \newcommand{\Bold}[1]{\mathbf{#1}}\left(\frac{{\left(\cos\left(\theta\right) \sin\left(\phi\right) + \cos\left(\phi\right) \sin\left(\theta\right)\right)} M_{2} - M_{1} \sin\left(\phi\right)}{M_{2}}, \left[\sin\left(\phi\right) = -\frac{M_{2} \cos\left(\phi\right) \sin\left(\theta\right)}{M_{2} \cos\left(\theta\right) - M_{1}}\right]\right) $$
Hence, the solution for $\phi$ is obtianed by intersecting the tangent function with a constant line, and by periodicity we can reason in the interval $[-\pi, \pi]$, recall that:
Remark. I'd be interested to know if there is a way to do this sort of "case study" automatically. I tried with some solver options (explicit, etc.), but it doesn't help. Neither with SymPy solve or the new solveset.. but maybe I'm missing to pass some extra information.
![]() | 12 | No.12 Revision |
It seems to me that the hard work is already done properly by the solver. Finally, you got:
sage: var("theta phi M_1 M_2")
sage: f = (sin(theta+phi)-M_1/M_2*sin(phi)).simplify_full()
sage: f, solve(f, phi)
Which is: $$ \newcommand{\Bold}[1]{\mathbf{#1}}\left(\frac{{\left(\cos\left(\theta\right) \sin\left(\phi\right) + \cos\left(\phi\right) \sin\left(\theta\right)\right)} M_{2} - M_{1} \sin\left(\phi\right)}{M_{2}}, \left[\sin\left(\phi\right) = -\frac{M_{2} \cos\left(\phi\right) \sin\left(\theta\right)}{M_{2} \cos\left(\theta\right) - M_{1}}\right]\right) $$
Hence, the solution for $\phi$ is obtianed by intersecting the tangent function with a constant line, and by periodicity we can reason in the interval $[-\pi, \pi]$, recall that:
Remark. I'd be interested to know if there is a way to do this sort of "case study" automatically. I tried with some solver options (explicit, etc.), but it doesn't help. Neither with SymPy solve or the new solveset.. but maybe I'm missing to pass some extra information.
![]() | 13 | No.13 Revision |
It seems to me that the hard work is already done properly by the solver. Finally, you got:
sage: var("theta phi M_1 M_2")
sage: f = (sin(theta+phi)-M_1/M_2*sin(phi)).simplify_full()
sage: f, solve(f, phi)
Which is: $$ \newcommand{\Bold}[1]{\mathbf{#1}}\left(\frac{{\left(\cos\left(\theta\right) \sin\left(\phi\right) + \cos\left(\phi\right) \sin\left(\theta\right)\right)} M_{2} - M_{1} \sin\left(\phi\right)}{M_{2}}, \left[\sin\left(\phi\right) = -\frac{M_{2} \cos\left(\phi\right) \sin\left(\theta\right)}{M_{2} \cos\left(\theta\right) - M_{1}}\right]\right) $$
Hence, the solution for $\phi$ is obtianed by intersecting the tangent function with a constant line, and by periodicity we can reason in the interval $[-\pi, \pi]$, recall that:
Remark. I'd be interested to know if there is a way to do this sort of "case study" automatically. I tried with some solver options (explicit, etc.), but it doesn't help. Neither with SymPy solve or the new solveset.. but maybe I'm missing to pass some extra information.
![]() | 14 | No.14 Revision |
It seems to me that the hard work is already done properly by the solver. Finally, you got:
sage: var("theta phi M_1 M_2")
sage: f = (sin(theta+phi)-M_1/M_2*sin(phi)).simplify_full()
sage: f, solve(f, phi)
Which is: $$ \newcommand{\Bold}[1]{\mathbf{#1}}\left(\frac{{\left(\cos\left(\theta\right) \sin\left(\phi\right) + \cos\left(\phi\right) \sin\left(\theta\right)\right)} M_{2} - M_{1} \sin\left(\phi\right)}{M_{2}}, \left[\sin\left(\phi\right) = -\frac{M_{2} \cos\left(\phi\right) \sin\left(\theta\right)}{M_{2} \cos\left(\theta\right) - M_{1}}\right]\right) $$
Hence, the solution for $\phi$ is obtianed by intersecting the tangent function with a constant line, and by periodicity we can reason in the interval $[-\pi, \pi]$, recall that:
Remark. I'd be interested to know if there is a way to do this sort of "case study" automatically. I tried with some solver options (explicit, etc.), but it doesn't help. Neither with SymPy solve or the new solveset.. but maybe I'm missing to pass some extra information.