To solve for the angle \( C \) using the inverse tangent function, we first need to identify what the sides represent. Let's assume:
- The side \( A = 15 \) is opposite to angle \( A \),
- The side \( B = 12 \) is opposite to angle \( B \),
- The side \( C = 9 \) is opposite to angle \( C \).
If we are looking specifically for the angle \( C \), we can use the tangent ratio. The tangent of an angle in a right triangle is the ratio of the lengths of the opposite side to the adjacent side. Assuming \( C \) is opposite to side \( C \) and angles \( A \) and \( B \) dictate the relationships, we can use:
\[ \tan(C) = \frac{\text{opposite}}{\text{adjacent}} \]
However, we need to clarify the adjacent side for vertex \( C \). Given that \( A \) and \( B \) are related, we typically use:
\[ \tan(C) = \frac{a}{b} \] where \( a \) is the length of the side opposite angle \( C \) and \( b \) is the length of the side adjacent to angle \( C \).
If we assume that we are to use \( A \) and \( B \) since they’re already known,
Let’s assume:
\[ \tan(C) = \frac{15}{12} \text{ (if we assume side A is opposite to angle A)} \]
Now we compute for \( C \):
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First calculate \( \tan^{-1}( \frac{15}{12} ) \):
\[ C \approx \tan^{-1} \left( \frac{15}{12} \right) \]
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Calculating:
\[ \frac{15}{12} = 1.25 \]
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Using a calculator or a table to find \( \tan^{-1}(1.25) \):
\[ C \approx 51.34° \]
Since we require rounding to the nearest whole degree, we would round \( 51.34° \) to \( 51° \).
However, since that option isn’t given, we would best choose the next closest, which reflects our situation. Considering the options:
- 37°
- 59°
- 39°
- 53°
The closest angle measured seems to be 53° considering rounding.
Thus the answer is 53°.
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