Find ` theta , " if " sin 5 theta = cos 4 theta : ` where ` 5 theta and 4 theta ` are acute angles.

Answer

To solve the equation \( \sin 5\theta = \cos 4\theta \) where \( 5\theta \) and \( 4\theta \) are acute angles, we can follow these steps:

### Step 1: Use the co-function identity
We know that \( \cos x = \sin(90^\circ - x) \). Therefore, we can rewrite \( \cos 4\theta \) as:
\[ \cos 4\theta = \sin(90^\circ - 4\theta) \]

### Step 2: Set up the equation
Now, substituting this into our original equation, we have:
\[ \sin 5\theta = \sin(90^\circ - 4\theta) \]

### Step 3: Use the property of sine
Since \( \sin A = \sin B \) implies that \( A = B \) or \( A = 180^\circ - B \), we can set up two equations:
1. \( 5\theta = 90^\circ - 4\theta \)
2. \( 5\theta = 180^\circ - (90^\circ - 4\theta) \)

### Step 4: Solve the first equation
From the first equation:
\[ 5\theta + 4\theta = 90^\circ \]
\[ 9\theta = 90^\circ \]
\[ \theta = \frac{90^\circ}{9} = 10^\circ \]

### Step 5: Solve the second equation
Now, solving the second equation:
\[ 5\theta = 180^\circ - 90^\circ + 4\theta \]
\[ 5\theta = 90^\circ + 4\theta \]
Subtracting \( 4\theta \) from both sides:
\[ 5\theta - 4\theta = 90^\circ \]
\[ \theta = 90^\circ \]
However, \( 90^\circ \) is not an acute angle, so we discard this solution.

### Conclusion
Thus, the only solution for \( \theta \) that satisfies the given conditions is:
\[ \theta = 10^\circ \]