The problem in the image asks you to find the value of a trigonometric expression, given that $3 \tan A = 4$. The expression to be evaluated is:

$\frac{3 \sin A + 4 \cos A}{6 \sin A - 5 \cos A}$

Step 1: Solving for $\tan A$

From the equation $3 \tan A = 4$, we can solve for $\tan A$: $\tan A = \frac{4}{3}$

Now, since $\tan A = \frac{\sin A}{\cos A}$, we have: $\frac{\sin A}{\cos A} = \frac{4}{3}$

Step 2: Expressing $\sin A$ and $\cos A$

Let $\sin A = 4k$ and $\cos A = 3k$, where $k$ is a constant. This assumption comes from the ratio $\frac{4}{3}$ derived from $\tan A$.

Using the Pythagorean identity $\sin^2 A + \cos^2 A = 1$, we can substitute these values: $(4k)^2 + (3k)^2 = 1$ $16k^2 + 9k^2 = 1$ $25k^2 = 1$ $k^2 = \frac{1}{25}$ $k = \frac{1}{5}$

Thus, we have: $\sin A = \frac{4}{5}, \quad \cos A = \frac{3}{5}$

Step 3: Substituting into the expression

Now substitute $\sin A = \frac{4}{5}$ and $\cos A = \frac{3}{5}$ into the expression: $\frac{3 \sin A + 4 \cos A}{6 \sin A - 5 \cos A}$ $= \frac{3 \times \frac{4}{5} + 4 \times \frac{3}{5}}{6 \times \frac{4}{5} - 5 \times \frac{3}{5}}$ $= \frac{\frac{12}{5} + \frac{12}{5}}{\frac{24}{5} - \frac{15}{5}}$ $= \frac{\frac{24}{5}}{\frac{9}{5}} = \frac{24}{9} = \frac{8}{3}$

Final Answer:

The value of the expression is $\frac{8}{3}$.

Would you like more details or have any further questions?

Here are some related questions you can explore:

1. How do you simplify trigonometric expressions using identities?
2. Can you derive values of other trigonometric functions given $\tan A$?
3. What are other methods to solve trigonometric expressions involving $\tan A$?
4. How can you generalize this approach for different angles?
5. What is the geometric interpretation of the problem?

Tip: Always use Pythagorean identities when working with both sine and cosine to simplify expressions easily.