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The problem is asking to find the trajectory of the resulting oscillation and the direction of motion for a point that participates in two mutually perpendicular oscillations described by:

\[
x = 5 \cos(\pi t) \, \text{(cm)}
\]
\[
y = 3 \sin(\pi t) \, \text{(cm)}
\]

### Solution:

To find the trajectory, we need to eliminate the parameter \( t \) from the equations to express \( y \) as a function of \( x \).

1. **Express \(\cos(\pi t)\) and \(\sin(\pi t)\) in terms of \( x \) and \( y \):**

   From \( x = 5 \cos(\pi t) \), we get:
   \[
   \cos(\pi t) = \frac{x}{5}
   \]

   From \( y = 3 \sin(\pi t) \), we get:
   \[
   \sin(\pit) = \frac{y}{3}
   \]

2. **Use the identity \(\sin^2(\pi t) + \cos^2(\pi t) = 1\):**

   Substitute the expressions for \(\cos(\pi t)\) and \(\sin(\pi t)\):
   \[
   \left(\frac{x}{5}\right)^2 + \left(\frac{y}{3}\right)^2 = 1
   \]

3. **Simplify the equation:**

   \[
   \frac{x^2}{25} + \frac{y^2}{9} = 1
   \]

This is the equation of an ellipse, indicating that the point moves along an elliptical trajectory.

### Direction of Motion

To determine the direction of motion, observe the time-dependent behavior of \( x \) and \( y \):
- As \( t \) increases from \( 0 \):
  - \( x = 5 \cos(\pi t) \) starts at \( 5 \) and oscillates between \( -5 \) and \( 5 \).
  - \( y = 3 \sin(\pi t) \) starts at \( 0 \) and oscillates between \( -3 \) and \( 3 \).

Since \(\cos(\pi t)\) and \(\sin(\pi t)\) indicate a counterclockwise motion on the \( x \)-\( y \) plane, the point moves counterclockwise along the ellipse.

### Final Answer
- **Trajectory:** Ellipse, given by \(\frac{x^2}{25} + \frac{y^2}{9} = 1\).
- **Direction of Motion:** Counterclockwise.

Would you like further details on any step?

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**Exploratory Questions:**

1. How would the trajectory change if the coefficients of \( \cos \) and \( \sin \) were different?
2. What if the oscillations had different frequencies?
3. Can you determine the maximum velocity of the point on this trajectory?
4. How would adding a phase shift to either \( x \) or \( y \) affect the trajectory?
5. Could we represent this motion in polar coordinates?

**Tip:** For oscillatory motion in two dimensions, always check if identities like \( \sin^2 + \cos^2 = 1 \) can simplify the expressions.

Точка бере участь у двох взаємно перпендикулярних коливаннях x = 5 cos(πt) (см) та y = 3 sin(πt) (см). Знайти траєкторію результуючого коливання та напрямок руху.

Solve a problem involving a point in two mutually perpendicular oscillations, x = 5cos(πt) and y = 3sin(πt), to find the resulting trajectory and direction of motion. This solution covers parametric equations and uses the Pythagorean identity to show the trajectory as an ellipse. Suitable for undergraduate students studying oscillations and parametric motion.

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