If `P sin theta = sqrt(3)` and `P costheta =1`, then the value of P is:

Options

A. `1/2`
B. `2/sqrt(3)`
C. `-1/sqrt(3)`
D. 2

Answer

To find the value of \( P \) given the equations \( P \sin \theta = \sqrt{3} \) and \( P \cos \theta = 1 \), we can follow these steps:

### Step 1: Write down the equations
We have:
1. \( P \sin \theta = \sqrt{3} \) (Equation 1)
2. \( P \cos \theta = 1 \) (Equation 2)

### Step 2: Square both equations
Now, we will square both equations to eliminate \( P \):
- Squaring Equation 1:
\[ (P \sin \theta)^2 = (\sqrt{3})^2 \]
This gives us:
\[ P^2 \sin^2 \theta = 3 \quad \text{(Equation 3)} \]

- Squaring Equation 2:
\[ (P \cos \theta)^2 = 1^2 \]
This gives us:
\[ P^2 \cos^2 \theta = 1 \quad \text{(Equation 4)} \]

### Step 3: Add the squared equations
Now, we add Equation 3 and Equation 4:
\[ P^2 \sin^2 \theta + P^2 \cos^2 \theta = 3 + 1 \]
This simplifies to:
\[ P^2 (\sin^2 \theta + \cos^2 \theta) = 4 \]

### Step 4: Use the Pythagorean identity
Using the Pythagorean identity, we know that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting this into our equation gives:
\[ P^2 \cdot 1 = 4 \]
Thus, we have:
\[ P^2 = 4 \]

### Step 5: Solve for \( P \)
Taking the square root of both sides:
\[ P = \sqrt{4} = 2 \]

### Final Answer
The value of \( P \) is \( 2 \).

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