Find the maximum value of 5 tan 8 cos 8 + 4 cos 8.

Answer

To find the maximum value of the expression \( 5 \tan \theta \cos \theta + 4 \cos \theta \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ y = 5 \tan \theta \cos \theta + 4 \cos \theta \]
We know that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Therefore, we can rewrite \( y \) as:
\[ y = 5 \frac{\sin \theta}{\cos \theta} \cos \theta + 4 \cos \theta \]
This simplifies to:
\[ y = 5 \sin \theta + 4 \cos \theta \]

### Step 2: Find the maximum value of \( y \)
To find the maximum value of \( y = 5 \sin \theta + 4 \cos \theta \), we can use the formula for the maximum value of a linear combination of sine and cosine:
\[ R = \sqrt{a^2 + b^2} \]
where \( a = 5 \) and \( b = 4 \).

Calculating \( R \):
\[ R = \sqrt{5^2 + 4^2} = \sqrt{25 + 16} = \sqrt{41} \]

### Step 3: Maximum value of \( y \)
The maximum value of \( y \) occurs when:
\[ y_{\text{max}} = R = \sqrt{41} \]

### Conclusion
Thus, the maximum value of \( 5 \tan \theta \cos \theta + 4 \cos \theta \) is:
\[ \sqrt{41} \]