If tan 3A = cot `(A - 25^(@))`, 3A `lt 90^(@)`, then find the value of `angle`A.

Answer

To solve the equation \( \tan 3A = \cot (A - 25^\circ) \) with the condition \( 3A < 90^\circ \), we can follow these steps:

### Step 1: Use the cotangent identity
We know that \( \cot \theta = \tan (90^\circ - \theta) \). Therefore, we can rewrite the equation as:
\[ \tan 3A = \tan (90^\circ - (A - 25^\circ)) \]

### Step 2: Simplify the right side
Now, simplify the expression on the right side:
\[ \tan 3A = \tan (90^\circ - A + 25^\circ) = \tan (115^\circ - A) \]

### Step 3: Set the angles equal
Since the tangent function is periodic, we can set the angles equal to each other:
\[ 3A = 115^\circ - A \]

### Step 4: Solve for A
Now, we can solve for \( A \):
\[ 3A + A = 115^\circ \]
\[ 4A = 115^\circ \]
\[ A = \frac{115^\circ}{4} = 28.75^\circ \]

### Step 5: Round the answer
Rounding \( 28.75^\circ \) gives us approximately:
\[ A \approx 29^\circ \]

Thus, the value of angle \( A \) is \( 29^\circ \).

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