Let's compute the resultant force using the Trigonometric Method and Component/Analytical Method based on the information given. Since the image shows two forces $F = 450 \, \text{N}$ at angles $30^\circ$ and $45^\circ$ relative to the horizontal axis, we'll assume there are two forces acting on a body and we need to find the resultant force.

1. Trigonometric Method:

For two forces $F_1$ and $F_2$ making angles $\theta_1$ and $\theta_2$ with the horizontal, the resultant force $R$ can be calculated using the Law of Cosines:

$R = \sqrt{F_1^2 + F_2^2 + 2 \cdot F_1 \cdot F_2 \cdot \cos(\theta)}$

where $\theta$ is the angle between the two forces.

Given:

• $F_1 = 450 \, \text{N}$
• $F_2 = 450 \, \text{N}$
• $\theta = 45^\circ - 30^\circ = 15^\circ$

Substitute the values into the formula:

$R = \sqrt{450^2 + 450^2 + 2 \cdot 450 \cdot 450 \cdot \cos(15^\circ)}$

$R = \sqrt{202500 + 202500 + 2 \cdot 450 \cdot 450 \cdot 0.9659}$

$R = \sqrt{202500 + 202500 + 392122.5}$

$R = \sqrt{797122.5}$

$R \approx 892.83 \, \text{N}$

2. Component/Analytical Method:

Let's break each force into its $x$ and $y$ components.

For $F_1 = 450 \, \text{N}$ at $30^\circ$:

$F_{1x} = 450 \cdot \cos(30^\circ) = 450 \cdot 0.866 = 389.7 \, \text{N}$ $F_{1y} = 450 \cdot \sin(30^\circ) = 450 \cdot 0.5 = 225 \, \text{N}$

For $F_2 = 450 \, \text{N}$ at $45^\circ$:

$F_{2x} = 450 \cdot \cos(45^\circ) = 450 \cdot 0.7071 = 318.2 \, \text{N}$ $F_{2y} = 450 \cdot \sin(45^\circ) = 450 \cdot 0.7071 = 318.2 \, \text{N}$

Now sum the components:

$R_x = F_{1x} + F_{2x} = 389.7 + 318.2 = 707.9 \, \text{N}$ $R_y = F_{1y} + F_{2y} = 225 + 318.2 = 543.2 \, \text{N}$

The magnitude of the resultant force $R$ is:

$R = \sqrt{R_x^2 + R_y^2} = \sqrt{707.9^2 + 543.2^2} = \sqrt{501137.7 + 295055.1} = \sqrt{796192.8}$

$R \approx 892.29 \, \text{N}$

Finally, the direction $\theta_R$ of the resultant is:

$\theta_R = \tan^{-1}\left(\frac{R_y}{R_x}\right) = \tan^{-1}\left(\frac{543.2}{707.9}\right)$

$\theta_R \approx 37.99^\circ$

Conclusion:

The magnitude of the resultant force is approximately $892.3 \, \text{N}$, and it makes an angle of $38^\circ$ with the horizontal axis.

Would you like to go over the graphical method or have any further questions?

Relative Questions:

1. What happens if the angles between the forces increase or decrease?
2. How does the resultant change if the forces are not equal in magnitude?
3. Could you derive the trigonometric method formula using vector addition principles?
4. How does the component method help in solving more complex systems?
5. Can this method be applied to find the resultant of three or more forces?

Tip:

Always verify your angle measurements and trigonometric calculations, especially in vector addition problems, to ensure accuracy in determining the resultant force.