Trigonometric Problem Analysis

The question asks us to find the value of a specific trigonometric expression, $\frac{8\sin A - 7\cos A}{8\sin A + 11\cos A}$, given a relationship involving the tangent function, specifically $8 \tan A = 5$. This type of problem often involves simplifying the expression using the given information.

Deriving tan A Value

First, let's find the value of $\tan A$ from the given equation:

$$8 \tan A = 5$$

Dividing both sides by 8, we get:

$$ \tan A = \frac{5}{8} $$

Simplifying Trigonometric Expression

The expression we need to evaluate is $\frac{8\sin A - 7\cos A}{8\sin A + 11\cos A}$. Notice that both the numerator and the denominator contain terms with $\sin A$ and $\cos A$. We know the relationship between $\tan A$, $\sin A$, and $\cos A$ is $\tan A = \frac{\sin A}{\cos A}$.

To express the given fraction in terms of $\tan A$, we can divide both the numerator and the denominator by $\cos A$ (assuming $\cos A \neq 0$).

$$ \frac{8\sin A - 7\cos A}{8\sin A + 11\cos A} = \frac{\frac{8\sin A - 7\cos A}{\cos A}}{\frac{8\sin A + 11\cos A}{\cos A}} $$

Now, distribute the division by $\cos A$ to each term:

$$ = \frac{\frac{8\sin A}{\cos A} - \frac{7\cos A}{\cos A}}{\frac{8\sin A}{\cos A} + \frac{11\cos A}{\cos A}} $$

Using the identity $\tan A = \frac{\sin A}{\cos A}$, we can simplify this to:

$$ = \frac{8\tan A - 7}{8\tan A + 11} $$

Calculating Final Trigonometric Value

Now we substitute the value of $\tan A = \frac{5}{8}$ into the simplified expression:

$$ \frac{8 \left( \frac{5}{8} \right) - 7}{8 \left( \frac{5}{8} \right) + 11} $$

Perform the multiplication:

$$ = \frac{5 - 7}{5 + 11} $$

Now, perform the subtraction and addition:

$$ = \frac{-2}{16} $$

Finally, simplify the fraction:

$$ = -\frac{1}{8} $$

Final Trigonometric Result

The value of the expression $\frac{8\sin A - 7\cos A}{8\sin A + 11\cos A}$ when $8 \tan A = 5$ is $-\frac{1}{8}$.