If `cos theta = (12)/(13) ` , find the value of `2 sin theta - 4 tan, theta " where " theta` is a acute

Answer

To solve the problem, we need to find the value of \(2 \sin \theta - 4 \tan \theta\) given that \(\cos \theta = \frac{12}{13}\) and \(\theta\) is an acute angle.

### Step 1: Understand the relationship between sides in a right triangle
Given \(\cos \theta = \frac{12}{13}\), we can interpret this in a right triangle where:
- The adjacent side (base) = 12
- The hypotenuse = 13

### Step 2: Find the length of the opposite side using the Pythagorean theorem
Using the Pythagorean theorem:
\[ AB^2 + BC^2 = AC^2 \]
Let \(AB\) be the opposite side, \(BC\) be the adjacent side (12), and \(AC\) be the hypotenuse (13).
\[ AB^2 + 12^2 = 13^2 \]
\[ AB^2 + 144 = 169 \]
\[ AB^2 = 169 - 144 = 25 \]
\[ AB = \sqrt{25} = 5 \]

### Step 3: Calculate \(\sin \theta\) and \(\tan \theta\)
Now we can find \(\sin \theta\) and \(\tan \theta\):
- \(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{5}{13}\)
- \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{5}{12}\)

### Step 4: Substitute values into the expression \(2 \sin \theta - 4 \tan \theta\)
Now substitute \(\sin \theta\) and \(\tan \theta\) into the expression:
\[ 2 \sin \theta - 4 \tan \theta = 2 \left(\frac{5}{13}\right) - 4 \left(\frac{5}{12}\right) \]
Calculating each term:
\[ = \frac{10}{13} - \frac{20}{12} \]

### Step 5: Find a common denominator and simplify
The common denominator of 13 and 12 is \(13 \times 12 = 156\):
\[ = \frac{10 \times 12}{156} - \frac{20 \times 13}{156} = \frac{120}{156} - \frac{260}{156} \]
\[ = \frac{120 - 260}{156} = \frac{-140}{156} \]

### Step 6: Simplify the fraction
Now simplify \(\frac{-140}{156}\):
\[ = \frac{-70}{78} = \frac{-35}{39} \]

### Final Answer
Thus, the value of \(2 \sin \theta - 4 \tan \theta\) is:
\[ \boxed{\frac{-35}{39}} \]