Thank you for the correction! Let's solve the problem with the correct information that \(\sin \theta = -\frac{5}{13}\).

### Updated Problem Summary:
We are given that \(\sin \theta = -\frac{5}{13}\) with \(\theta\) in the fourth quadrant. We need to evaluate the expression:

\[
\sec \theta + 26 \sin \theta - 4 \tan \theta
\]

### Step 1: Determine the other trigonometric values
Given:
\[
\sin \theta = -\frac{5}{13}
\]

Since \(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}\), the opposite side is \(-5\) and the hypotenuse is \(13\). To find the adjacent side \(x\), we use the Pythagorean theorem:

\[
x^2 + (-5)^2 = 13^2
\]
\[
x^2 + 25 = 169
\]
\[
x^2 = 144 \implies x = 12
\]

Since \(\theta\) is in the fourth quadrant, the adjacent side \(x\) is positive. So, \(x = 12\).

Now, we can find the other trigonometric ratios:
\[
\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{12}{13}
\]
\[
\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{-5}{12}
\]

### Step 2: Evaluate the required expression
We now substitute these values into the expression:

1. **Secant:**
   \[
   \sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{12}{13}} = \frac{13}{12}
   \]

2. **Sin:**
   \[
   26 \sin \theta = 26 \times \left(-\frac{5}{13}\right) = -10
   \]

3. **Tangent:**
   \[
   4 \tan \theta = 4 \times \left(\frac{-5}{12}\right) = -\frac{20}{12} = -\frac{5}{3}
   \]

### Step 3: Combine all the terms
Now, let's add them together:
\[
\sec \theta + 26 \sin \theta - 4 \tan \theta = \frac{13}{12} + (-10) - \left(-\frac{5}{3}\right)
\]

Convert \(-10\) into a fraction with denominator 12:
\[
-10 = -\frac{120}{12}
\]

Now, the expression is:
\[
\frac{13}{12} - \frac{120}{12} + \frac{20}{12} = \frac{13 - 120 + 20}{12} = \frac{-87}{12}
\]

Simplify the fraction:
\[
\frac{-87}{12} = -\frac{29}{4}
\]

### Final Answer:
\[
\sec \theta + 26 \sin \theta - 4 \tan \theta = -\frac{29}{4}
\]

Would you like any additional explanations or have any further questions?

Here are some related questions that might interest you:
1. Why is it important to determine the correct quadrant when evaluating trigonometric functions?
2. How do you simplify trigonometric expressions involving multiple operations?
3. What are the typical steps to verify if your trigonometric identities are correct?
4. How can the Pythagorean theorem be applied in different contexts?
5. Can you explain the relationship between \(\sin \theta\) and \(\cos \theta\) in different quadrants?

**Tip:** When dealing with trigonometric functions, always make sure to account for the signs based on the quadrant in which the angle lies.

Learn how to evaluate the expression sec theta + 26 sin theta - 4 tan theta with sin theta = -5/13 in the fourth quadrant. Detailed step-by-step solution including trigonometric values and operations. Suitable for Grades 10-12 students studying trigonometry.

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