To use the Law of Sines to find side \( a \) in the triangle where \( m\angle C = 38° \) and side \( c = 22 \text{ mi} \), we start by noting that to apply the Law of Sines, we typically look for the relationship between the sine of the angles and the lengths of the opposite sides.
In this case, we also need to know the angle opposite to side \( a \). However, since angle \( C = 38° \) is given, we can deduce that angle \( A + B + C = 180° \). Assuming that we can derive or are given another angle (or can deduce it), then we would set up the Law of Sines as follows:
\[ \frac{a}{\sin A} = \frac{c}{\sin C} \]
This yields:
\[ \frac{a}{\sin A} = \frac{22 \text{ mi}}{\sin 38°} \]
From this equation, we can isolate \( a \):
\[ a = \frac{22 \text{ mi} \cdot \sin A}{\sin 38°} \]
However, based on the original question's options, we can find the correct format to represent the Law of Sines based on the responses listed.
Considering the responses you provided, the correct equation based on \( m∠C = 38° \) and side \( c = 22 \text{ mi} \) would involve \( \sin 38° \) with \( a \):
Thus, the answer can be represented as:
\[ \frac{\sin 38°}{22 \text{ mi}} = \frac{\sin A}{a} \]
This suggests that:
The correct response should be:
\(\sin 38° / 22 mi. = \sin 90° / a\)
This implies that the angle \( A \) being opposite to side \( a \) must be \( 90° \), meaning we are dealing with a right triangle for this particular option.
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