given: F: 450 lb at 45 degrees (north west), Q: 200lb at 30 degrees (north east) , compute for the resultant using trigonometric method and analytical method

### Trigonometric Method

To find the resultant of two forces using the trigonometric method, we can use the Law of Cosines and Law of Sines. The problem involves adding two forces \( \mathbf{F} \) and \( \mathbf{Q} \) that are not aligned but have a certain angle between them.

#### Step 1: Break down the forces into their components
Given:
- \( \mathbf{F} = 450 \, \text{lb} \) at \( 45^\circ \) (North-West)
- \( \mathbf{Q} = 200 \, \text{lb} \) at \( 30^\circ \) (North-East)

Let's first convert the angles relative to the positive x-axis (East direction):
- For \( \mathbf{F} \): The angle from the positive x-axis is \( 180^\circ - 45^\circ = 135^\circ \).
- For \( \mathbf{Q} \): The angle from the positive x-axis is \( 30^\circ \).

#### Step 2: Calculate the angle between the two forces
The angle between \( \mathbf{F} \) and \( \mathbf{Q} \) is:
\[ \theta = 135^\circ - 30^\circ = 105^\circ \]

#### Step 3: Apply the Law of Cosines
To find the magnitude of the resultant \( \mathbf{R} \):
\[ R = \sqrt{F^2 + Q^2 + 2 \cdot F \cdot Q \cdot \cos(\theta)} \]

Substitute the values:
\[ R = \sqrt{450^2 + 200^2 + 2 \cdot 450 \cdot 200 \cdot \cos(105^\circ)} \]

Let's calculate this.

#### Step 4: Calculate the angle of the resultant using Law of Sines
To find the direction (angle) of the resultant \( \mathbf{R} \), we use the Law of Sines:
\[ \frac{\sin(\alpha)}{Q} = \frac{\sin(\theta)}{R} \]

Where:
- \( \alpha \) is the angle between \( \mathbf{F} \) and \( \mathbf{R} \).
  
Solve for \( \alpha \), and then find the actual angle \( \beta \) of \( \mathbf{R} \) with respect to the positive x-axis (East):
\[ \beta = \alpha + \text{(angle of } \mathbf{Q} \text{ relative to the x-axis)} \]

### Analytical Method (Component Method)

#### Step 1: Resolve the forces into x and y components
For \( \mathbf{F} \) at \( 135^\circ \):
- \( F_x = 450 \cdot \cos(135^\circ) \)
- \( F_y = 450 \cdot \sin(135^\circ) \)

For \( \mathbf{Q} \) at \( 30^\circ \):
- \( Q_x = 200 \cdot \cos(30^\circ) \)
- \( Q_y = 200 \cdot \sin(30^\circ) \)

#### Step 2: Sum the components to find the resultant
- \( R_x = F_x + Q_x \)
- \( R_y = F_y + Q_y \)

#### Step 3: Calculate the magnitude and direction of the resultant
Magnitude:
\[ R = \sqrt{R_x^2 + R_y^2} \]

Direction:
\[ \theta_R = \tan^{-1}\left(\frac{R_y}{R_x}\right) \]

Let me calculate these values for you.### Result

Both the trigonometric and analytical methods yield the same result:

- **Magnitude of the resultant \( R \)**: \( 442.62 \, \text{lb} \)
- **Direction of the resultant \( R \)**: \( 109.12^\circ \) relative to the positive x-axis (East direction)

This angle corresponds to a direction that is slightly south of due west, confirming the directionality influenced by the forces' orientation.

Would you like more details, or have any further questions?

Here are 5 related questions you might explore:
1. How does the resultant change if the angle between the two forces is reduced to 90 degrees?
2. What happens to the resultant if both forces are applied in the same direction?
3. How would you find the resultant if one of the forces were zero?
4. How can you verify the direction of the resultant graphically?
5. What is the significance of the Law of Sines in finding the angle of the resultant?

**Tip:** When working with vector components, always ensure the angles are measured from the same reference direction to maintain consistency in your calculations.

Learn how to compute the resultant of forces using both trigonometric and analytical methods. This problem involves vector addition, trigonometric calculations using the Law of Cosines and Law of Sines, and analytical methods using component resolution. Get detailed step-by-step solutions and calculations.

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