https://r2.erweima.ai/mathbot/image/17261639179736pyy028p.jpg

To solve this problem, we need to determine the magnitudes of forces \( F_1 \) and \( F_2 \) such that the particle is in equilibrium. 

### Given:
- A force of \( 12 \, \text{N} \) acting along the negative x-axis.
- Force \( F_1 \) at an angle of \( 45^\circ \) with the positive x-axis.
- Force \( F_2 \) at an angle of \( 30^\circ \) with the negative x-axis.

### Equilibrium Conditions:
For the particle to be in equilibrium, the sum of forces in both the x and y directions must be zero.

1. **Sum of forces in the x-direction:**

\[
\sum F_x = 0 \implies F_1 \cos(45^\circ) - F_2 \cos(30^\circ) - 12 = 0
\]

2. **Sum of forces in the y-direction:**

\[
\sum F_y = 0 \implies F_1 \sin(45^\circ) + F_2 \sin(30^\circ) = 0
\]

### Calculations:

Let's calculate the components:

- \( \cos(45^\circ) = \sin(45^\circ) = \frac{\sqrt{2}}{2} \)
- \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \), \( \sin(30^\circ) = \frac{1}{2} \)

Substitute these values into the equations:

1. **For the x-components:**

\[
F_1 \frac{\sqrt{2}}{2} - F_2 \frac{\sqrt{3}}{2} = 12
\]

2. **For the y-components:**

\[
F_1 \frac{\sqrt{2}}{2} + F_2 \frac{1}{2} = 0
\]

Solve these equations simultaneously to find \( F_1 \) and \( F_2 \).

Let's start by solving for \( F_1 \) and \( F_2 \).

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

Learn how to determine the magnitudes of F1 and F2 for a particle in equilibrium using trigonometric principles. This statics problem covers key concepts of vectors, equilibrium, and force decomposition.

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