In a right triangle, the tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side. In this case, you want to find the length of side AB (the side opposite angle C) using the given information.
Given:
- Angle C = 26 degrees
- Side BC (adjacent to angle C) = 15 inches
- Angle B = 90 degrees
The tangent ratio can be expressed as:
\[ \tan(C) = \frac{\text{opposite}}{\text{adjacent}} \]
Substituting the known values:
\[ \tan(26^\circ) = \frac{AB}{BC} \]
where:
- \( AB \) is the opposite side,
- \( BC \) is the adjacent side (15 inches).
Now plug in the values:
\[ \tan(26^\circ) = \frac{AB}{15} \]
To find \( AB \), we can rearrange this equation to:
\[ AB = 15 \cdot \tan(26^\circ) \]
Next, we can use a calculator to find \( \tan(26^\circ) \):
\[ \tan(26^\circ) \approx 0.4877 \]
Now calculate \( AB \):
\[ AB \approx 15 \cdot 0.4877 \approx 7.315 \]
Rounded to the nearest hundredths, the length of AB is:
\[ \boxed{7.32} \text{ inches} \]
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