If `4 tan theta = 3,` evaluate `((4 sin theta -2 cos theta+3)/(4 sin theta + 2cos theta-5))`

Answer

To solve the problem, we will follow these steps:

### Step 1: Find the value of \(\tan \theta\)
Given that \(4 \tan \theta = 3\), we can isolate \(\tan \theta\):
\[ \tan \theta = \frac{3}{4} \]

### Step 2: Set up a right triangle
In a right triangle, \(\tan \theta\) is defined as the ratio of the opposite side (perpendicular) to the adjacent side (base). Therefore, we can denote:
- Opposite side = 3 (perpendicular)
- Adjacent side = 4 (base)

### Step 3: Calculate the hypotenuse using the Pythagorean theorem
Using the Pythagorean theorem:
\[ h^2 = p^2 + b^2 \]
Substituting the values we have:
\[ h^2 = 3^2 + 4^2 = 9 + 16 = 25 \]
Thus, the hypotenuse \(h\) is:
\[ h = \sqrt{25} = 5 \]

### Step 4: Find \(\sin \theta\) and \(\cos \theta\)
Now we can find \(\sin \theta\) and \(\cos \theta\):
\[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{5} \]
\[ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{4}{5} \]

### Step 5: Substitute \(\sin \theta\) and \(\cos \theta\) into the expression
We need to evaluate:
\[ \frac{4 \sin \theta - 2 \cos \theta + 3}{4 \sin \theta + 2 \cos \theta - 5} \]
Substituting the values of \(\sin \theta\) and \(\cos \theta\):
\[ = \frac{4 \left(\frac{3}{5}\right) - 2 \left(\frac{4}{5}\right) + 3}{4 \left(\frac{3}{5}\right) + 2 \left(\frac{4}{5}\right) - 5} \]

### Step 6: Simplify the numerator
Calculating the numerator:
\[ = \frac{\frac{12}{5} - \frac{8}{5} + 3}{\frac{12}{5} + \frac{8}{5} - 5} \]
Combining the terms:
\[ = \frac{\frac{12 - 8 + 15}{5}}{\frac{12 + 8 - 25}{5}} = \frac{\frac{19}{5}}{\frac{-5}{5}} = \frac{19}{-5} = -\frac{19}{5} \]

### Step 7: Final answer
Thus, the value of the expression is:
\[ -\frac{19}{5} \]

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