Find the value of x if tan 3 x = sin `45^(@) cos 45^(@) + sin 30^(@)`

Answer

To solve the equation \( \tan 3x = \sin 45^\circ \cos 45^\circ + \sin 30^\circ \), we can follow these steps:

### Step 1: Calculate \( \sin 45^\circ \) and \( \cos 45^\circ \)
We know that:
\[ \sin 45^\circ = \frac{1}{\sqrt{2}} \quad \text{and} \quad \cos 45^\circ = \frac{1}{\sqrt{2}} \]

### Step 2: Calculate \( \sin 30^\circ \)
We also know that:
\[ \sin 30^\circ = \frac{1}{2} \]

### Step 3: Substitute these values into the equation
Now substitute the values into the equation:
\[ \tan 3x = \sin 45^\circ \cos 45^\circ + \sin 30^\circ \]
This becomes:
\[ \tan 3x = \left(\frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}}\right) + \frac{1}{2} \]

### Step 4: Simplify the right-hand side
Calculating the product:
\[ \frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}} = \frac{1}{2} \]
Now adding \( \frac{1}{2} \):
\[ \tan 3x = \frac{1}{2} + \frac{1}{2} = 1 \]

### Step 5: Set \( \tan 3x \) equal to \( 1 \)
We know that:
\[ \tan 45^\circ = 1 \]
Thus, we can write:
\[ \tan 3x = \tan 45^\circ \]

### Step 6: Solve for \( 3x \)
Since the tangents are equal, we can set their angles equal:
\[ 3x = 45^\circ \]

### Step 7: Solve for \( x \)
Now, divide both sides by 3:
\[ x = \frac{45^\circ}{3} = 15^\circ \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{15^\circ} \]

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