General solution of intersection of sines

Question

Is there a general way to solve this equation for $x$:

$$ A_1 \sin ( B_1 x + C_1) = A_2 \sin ( B_2 x + C_2) $$

where $A_1, B_1, C_1, A_2, B_2, C_2$ and $x$ are real, and $A_1, B_1, A_2$ and $B_2$ are not zero?

Answer 1

Since the two sine waves are out of phase, I don't think this will be possible, in general. I plotted an example:

import numpy as np
import matplotlib.pyplot as plt

t = np.arange(0.00, 20.0, 0.1)
A1 = np.sqrt(2)
A2 = np.sqrt(3)
B1 = np.sqrt(5)
B2 = np.sqrt(7)
C1 = np.sqrt(11)
C2 = np.sqrt(13)

data1 = A1*(np.sin(B1*t +C1))
data2 = A2*(np.sin(B2*t +C2))

fig, ax1 = plt.subplots()
color = 'tab:red'
ax1.plot(t, data1, color=color)
color = 'tab:blue'
ax1.plot(t, data2, color=color)

plt.show()

This gave me

It looks distinctly unpromising, at least to me.