To find the measure of angle \( C \) using the Law of Sines, we start with the relationship given by the Law of Sines:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
We know that:
- \( B = 94^\circ \)
- \( b = 13 \) inches
- \( c = 6 \) inches
We can first find the side length \( a \) using the formula:
\[ \frac{b}{\sin B} = \frac{c}{\sin C} \]
This allows us to isolate the sine of angle \( C \):
\[ \frac{c}{\sin C} = \frac{b}{\sin B} \]
Rearranging gives:
\[ \sin C = \frac{c \cdot \sin B}{b} \]
Plugging in the values we have:
\[ \sin C = \frac{6 \cdot \sin(94^\circ)}{13} \]
Now, we calculate \( \sin(94^\circ) \):
\[ \sin(94^\circ) \approx 0.998 \]
Now substituting this value in:
\[ \sin C \approx \frac{6 \cdot 0.998}{13} \]
Calculating this gives:
\[ \sin C \approx \frac{5.988}{13} \approx 0.4606 \]
Next, we find angle \( C \) by taking the inverse sine:
\[ C \approx \sin^{-1}(0.4606) \]
Using a calculator, we find:
\[ C \approx 27.4^\circ \]
Rounding this to the nearest degree gives:
\[ C \approx 27^\circ \]
Therefore, the measure of angle \( C \) is approximately 27 degrees.
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