Two forces of magnitudes P and Q are inclined at an angle `(theta)`. The magnitude of their resultant is 3Q. When the inclination is changed to `(180^(@)-theta)`, the magnitude of the resultant force becomes Q. Find the ratio of the forces.

Explanation

To solve the problem step by step, we will use the concepts of vector addition and the law of cosines.

### Step 1: Set Up the Problem
We have two forces, \( P \) and \( Q \), inclined at an angle \( \theta \). The magnitude of their resultant is given as \( R_1 = 3Q \).

### Step 2: Use the Law of Cosines for the First Case
The magnitude of the resultant \( R \) of two vectors can be calculated using the formula:
\[ R = \sqrt{P^2 + Q^2 + 2PQ \cos(\theta)} \]
Substituting \( R_1 = 3Q \):
\[ 3Q = \sqrt{P^2 + Q^2 + 2PQ \cos(\theta)} \]
Squaring both sides gives:
\[ (3Q)^2 = P^2 + Q^2 + 2PQ \cos(\theta) \]
\[ 9Q^2 = P^2 + Q^2 + 2PQ \cos(\theta) \]
Rearranging this, we get:
\[ P^2 + 2PQ \cos(\theta) = 8Q^2 \quad \text{(Equation 1)} \]

### Step 3: Use the Law of Cosines for the Second Case
When the angle is changed to \( 180^\circ - \theta \), the magnitude of the resultant becomes \( R_2 = Q \):
\[ R_2 = \sqrt{P^2 + Q^2 + 2PQ \cos(180^\circ - \theta)} \]
Since \( \cos(180^\circ - \theta) = -\cos(\theta) \), we have:
\[ Q = \sqrt{P^2 + Q^2 - 2PQ \cos(\theta)} \]
Squaring both sides gives:
\[ Q^2 = P^2 + Q^2 - 2PQ \cos(\theta) \]
Rearranging this, we get:
\[ P^2 - 2PQ \cos(\theta) = 0 \quad \text{(Equation 2)} \]

### Step 4: Solve the Equations
From Equation 2, we can express \( P^2 \) in terms of \( Q \):
\[ P^2 = 2PQ \cos(\theta) \]
Now substitute \( P^2 \) from Equation 2 into Equation 1:
\[ 2PQ \cos(\theta) + 2PQ \cos(\theta) = 8Q^2 \]
This simplifies to:
\[ 4PQ \cos(\theta) = 8Q^2 \]
Dividing both sides by \( 4Q \) (assuming \( Q \neq 0 \)):
\[ P \cos(\theta) = 2Q \]
Thus, we can express \( P \) in terms of \( Q \):
\[ P = \frac{2Q}{\cos(\theta)} \]

### Step 5: Find the Ratio of Forces
Now we can find the ratio \( \frac{P}{Q} \):
\[ \frac{P}{Q} = \frac{2}{\cos(\theta)} \]

### Step 6: Conclusion
The ratio of the magnitudes of the forces \( P \) and \( Q \) is:
\[ \frac{P}{Q} = 2 \]