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The image you provided shows a statics problem where two forces \( F_1 \) and \( F_2 \) are applied at angles to the horizontal axis, and the task is to determine their magnitudes such that the particle remains in equilibrium. 

### Problem Breakdown:
1. **Given Information:**
   - There is a force \( F_1 \) acting at an angle of \( 45^\circ \) above the horizontal (positive x-axis).
   - There is a force \( F_2 \) acting at an angle of \( 30^\circ \) below the horizontal (positive x-axis).
   - There is a horizontal force of \( 12 \, \text{N} \) acting to the left (along the negative x-axis).

2. **Objective:**
   - Determine the magnitudes of \( F_1 \) and \( F_2 \) such that the particle remains in equilibrium. For equilibrium, the sum of forces in both the x-direction and y-direction must equal zero.

### Steps to Solve:

#### Step 1: Resolve forces into their components
For both \( F_1 \) and \( F_2 \), resolve them into their horizontal (x) and vertical (y) components.

- \( F_1 \) components:
  - \( F_{1x} = F_1 \cos(45^\circ) \)
  - \( F_{1y} = F_1 \sin(45^\circ) \)

- \( F_2 \) components:
  - \( F_{2x} = F_2 \cos(30^\circ) \)
  - \( F_{2y} = -F_2 \sin(30^\circ) \) (since it is directed downward)

- The horizontal force is \( -12 \, \text{N} \) (acting to the left).

#### Step 2: Apply the equilibrium conditions
- For equilibrium in the x-direction (sum of forces in the x-direction must be zero):
  \[
  F_{1x} + F_{2x} - 12 = 0
  \]
  \[
  F_1 \cos(45^\circ) + F_2 \cos(30^\circ) = 12
  \]

- For

Determine os módulos de F1 e F2 de modo que a partícula fique em equilíbrio.

Solve a problem involving the equilibrium of forces where F1 and F2 are applied at specific angles. This problem covers statics, equilibrium conditions, and vector component calculations. Ideal for university-level students in engineering or physics.

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