To find the side length of \( AB \) in the right triangle given that angle \( C \) is 26 degrees and side \( BC \) (the side opposite angle \( C \)) is 15 inches, we can use trigonometric functions.
We will use the sine function, which relates the opposite side to the hypotenuse:
\[ \sin(C) = \frac{\text{opposite}}{\text{hypotenuse}} \]
In our triangle:
- \( C = 26^\circ \)
- \( \text{opposite} = BC = 15 \) inches
- \( AB \) is the hypotenuse.
Using the sine function:
\[ \sin(26^\circ) = \frac{15}{AB} \]
Now, rearranging to solve for \( AB \):
\[ AB = \frac{15}{\sin(26^\circ)} \]
Now we calculate \( \sin(26^\circ) \):
Using a calculator:
\[ \sin(26^\circ) \approx 0.43837 \]
Now substitute this value back into the equation:
\[ AB = \frac{15}{0.43837} \approx 34.1968 \]
Rounding to the nearest hundredth:
\[ AB \approx 34.20 \text{ inches} \]
Therefore, the side length of \( AB \) is approximately 34.20 inches.