If `xcos60^(@)+ycos0^(@)=3` and `4xsin30^(@)-ycot45^(@)=2`, then what is the value of `x` ?

Options

A. `-1`
B. `0`
C. `1`
D. `2`

Answer

To solve the given problem step by step, we will first rewrite the equations based on the trigonometric values provided and then solve for \( x \).

### Step 1: Write down the equations
We have two equations based on the problem statement:
1. \( x \cos 60^\circ + y \cos 0^\circ = 3 \)
2. \( 4x \sin 30^\circ - y \cot 45^\circ = 2 \)

### Step 2: Substitute the trigonometric values
We know the following values:
- \( \cos 60^\circ = \frac{1}{2} \)
- \( \cos 0^\circ = 1 \)
- \( \sin 30^\circ = \frac{1}{2} \)
- \( \cot 45^\circ = 1 \)

Substituting these values into the equations gives:
1. \( x \cdot \frac{1}{2} + y \cdot 1 = 3 \)
2. \( 4x \cdot \frac{1}{2} - y \cdot 1 = 2 \)

### Step 3: Simplify the equations
Now we simplify both equations:
1. \( \frac{x}{2} + y = 3 \)
2. \( 2x - y = 2 \)

### Step 4: Rearrange the first equation
From the first equation, we can express \( y \) in terms of \( x \):
\[ y = 3 - \frac{x}{2} \]

### Step 5: Substitute \( y \) in the second equation
Now substitute \( y \) in the second equation:
\[ 2x - (3 - \frac{x}{2}) = 2 \]

### Step 6: Simplify the equation
Distributing the negative sign gives:
\[ 2x - 3 + \frac{x}{2} = 2 \]
Now, combine like terms:
\[ 2x + \frac{x}{2} - 3 = 2 \]

### Step 7: Clear the fraction
To eliminate the fraction, multiply the entire equation by 2:
\[ 4x + x - 6 = 4 \]

### Step 8: Combine like terms
Combine \( 4x + x \):
\[ 5x - 6 = 4 \]

### Step 9: Solve for \( x \)
Add 6 to both sides:
\[ 5x = 10 \]
Now divide by 5:
\[ x = 2 \]

### Final Answer
Thus, the value of \( x \) is \( 2 \).

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