It seems that you're working with a fairly complex trigonometric expression that involves multiple terms with functions of $\omega t$, including sine and cosine, and some constants like $A$, $\gamma$, $k$, and $\phi$.

Let's break down the given equation step by step:

$- A m \omega^2 \sin(\omega t) \cos(\phi) + \cos(\omega t) \sin(\phi) + A \gamma \left( \cos(\omega t) \cos(\phi) - \sin(\omega t) \sin(\phi) \right) + A k \left( \sin(\omega t) \cos(\phi) + \cos(\omega t) \sin(\phi) \right) = a \sin(\omega t)$

1. Grouping the terms:

There are multiple terms involving sine and cosine functions of $\omega t$. To simplify, let's collect terms with $\sin(\omega t)$ and $\cos(\omega t)$ separately. We also have some constants and coefficients like $A$, $\gamma$, $k$, and $\phi$.

• Terms with $\sin(\omega t)$:

  • $- A m \omega^2 \sin(\omega t) \cos(\phi)$
  • $A k \sin(\omega t) \cos(\phi)$

• Terms with $\cos(\omega t)$:

  • $A \gamma \cos(\omega t) \cos(\phi)$
  • $\cos(\omega t) \sin(\phi)$

2. Simplifying each side:

• The term $A \gamma (\cos(\omega t) \cos(\phi) - \sin(\omega t) \sin(\phi))$ is a standard trigonometric identity: $\cos(A - B) = \cos(A)\cos(B) + \sin(A)\sin(B)$, so we can rewrite it as: $A \gamma \cos(\omega t - \phi)$

• Similarly, the term $A k (\sin(\omega t) \cos(\phi) + \cos(\omega t) \sin(\phi))$ can be rewritten as: $A k \sin(\omega t + \phi)$

3. Final rewritten equation:

Now the equation becomes:

$- A m \omega^2 \sin(\omega t) \cos(\phi) + \cos(\omega t) \sin(\phi) + A \gamma \cos(\omega t - \phi) + A k \sin(\omega t + \phi) = a \sin(\omega t)$

At this point, simplifying further requires using trigonometric identities, which might not immediately lead to a clean simplification. However, the equation is now broken down into more manageable parts.

Do you want further details on simplifying this equation or solving for any specific variable?

Related questions to explore:

1. What is the trigonometric identity used to simplify $A \gamma \cos(\omega t - \phi)$?
2. How can I solve this type of equation if I need to find $A$, $\gamma$, or other constants?
3. What happens if we apply a Fourier transform to this equation?
4. How does the solution of this equation change if $k = 0$?
5. Could this equation describe a physical system like a harmonic oscillator? If so, how would that affect the interpretation?

Tip: When simplifying complex trigonometric expressions, it's often helpful to first identify any standard identities that can reduce the complexity, such as $\sin(A + B)$ or $\cos(A - B)$.