`sin 5 theta = cos 2 theta , 0^(@) lt theta lt 180^(@).`
Find value of `theta`

Answer

To solve the equation \( \sin 5\theta = \cos 2\theta \) for \( 0^\circ < \theta < 180^\circ \), we can follow these steps:

### Step 1: Use the identity to convert sine to cosine
We know that:
\[ \sin x = \cos(90^\circ - x) \]
Using this identity, we can rewrite \( \sin 5\theta \) as:
\[ \sin 5\theta = \cos(90^\circ - 5\theta) \]
Thus, we can rewrite the original equation:
\[ \cos(90^\circ - 5\theta) = \cos 2\theta \]

### Step 2: Set up the cosine equation
From the equation \( \cos A = \cos B \), we have two cases:
1. \( A = B + 360^\circ n \)
2. \( A = -B + 360^\circ n \)

For our case, we will consider:
\[ 90^\circ - 5\theta = 2\theta \quad \text{(Case 1)} \]
and
\[ 90^\circ - 5\theta = -2\theta \quad \text{(Case 2)} \]

### Step 3: Solve Case 1
From Case 1:
\[ 90^\circ - 5\theta = 2\theta \]
Rearranging gives:
\[ 90^\circ = 7\theta \]
Thus,
\[ \theta = \frac{90^\circ}{7} \approx 12.8571^\circ \]

### Step 4: Solve Case 2
From Case 2:
\[ 90^\circ - 5\theta = -2\theta \]
Rearranging gives:
\[ 90^\circ = 3\theta \]
Thus,
\[ \theta = \frac{90^\circ}{3} = 30^\circ \]

### Step 5: Check the validity of solutions
Now we need to check if these values of \( \theta \) lie within the given range \( 0^\circ < \theta < 180^\circ \):
- \( \theta \approx 12.8571^\circ \) is valid.
- \( \theta = 30^\circ \) is also valid.

### Final Answer
The values of \( \theta \) that satisfy the equation \( \sin 5\theta = \cos 2\theta \) in the interval \( 0^\circ < \theta < 180^\circ \) are:
\[ \theta \approx 12.8571^\circ \quad \text{and} \quad \theta = 30^\circ \]