Question
In the triangle shown, m∠C=38° and c = 22 mi. Which equation shows how the Law of Sines can be used to find a? (1 point) Responses sin52°22 mi.=sin38°a the fraction with numerator sine 52 degrees and denominator 22 mi. is equal to the fraction with numerator sine 38 degrees and denominator A sin38°22 mi.=sin90°a the fraction with numerator sine 38 degrees and denominator 22 mi. is equal to the fraction with numerator sine 90 degrees and denominator A sin38°22 mi.=sin52°a the fraction with numerator sine 38 degrees and denominator 22 mi. is equal to the fraction with numerator sine 52 degrees and denominator A sin90°22 mi.=sin52°a
Answers
GPT-4o mini
To use the Law of Sines to find side \( a \) in the triangle, we need to relate the sides and angles of the triangle. The Law of Sines states that:
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]
In your case, we have:
- \( m\angle C = 38^\circ \)
- \( c = 22 \) mi
- \( m\angle A = 52^\circ \) (since \( A + B + C = 180^\circ \), and we can find \( B \) as \( 90^\circ - 38^\circ - 52^\circ \) which gives \( 90^\circ \))
Now, we'll use the Law of Sines in the form:
\[
\frac{a}{\sin A} = \frac{c}{\sin C}
\]
Substituting the known values:
\[
\frac{a}{\sin 52^\circ} = \frac{22 \text{ mi}}{\sin 38^\circ}
\]
Thus, rearranging this gives us:
\[
a = \frac{22 \text{ mi} \cdot \sin 52^\circ}{\sin 38^\circ}
\]
So, the correct equation showing how the Law of Sines can be used to find \( a \) is:
\[
\frac{\sin 52^\circ}{22 \text{ mi}} = \frac{\sin 38^\circ}{a}
\]
This corresponds to the response:
\( \sin 38^\circ \, 22 \text{ mi.} = \sin 52^\circ \, a \)
Therefore, the correct answer is:
**sin 38° 22 mi. = sin 52° a.**
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]
In your case, we have:
- \( m\angle C = 38^\circ \)
- \( c = 22 \) mi
- \( m\angle A = 52^\circ \) (since \( A + B + C = 180^\circ \), and we can find \( B \) as \( 90^\circ - 38^\circ - 52^\circ \) which gives \( 90^\circ \))
Now, we'll use the Law of Sines in the form:
\[
\frac{a}{\sin A} = \frac{c}{\sin C}
\]
Substituting the known values:
\[
\frac{a}{\sin 52^\circ} = \frac{22 \text{ mi}}{\sin 38^\circ}
\]
Thus, rearranging this gives us:
\[
a = \frac{22 \text{ mi} \cdot \sin 52^\circ}{\sin 38^\circ}
\]
So, the correct equation showing how the Law of Sines can be used to find \( a \) is:
\[
\frac{\sin 52^\circ}{22 \text{ mi}} = \frac{\sin 38^\circ}{a}
\]
This corresponds to the response:
\( \sin 38^\circ \, 22 \text{ mi.} = \sin 52^\circ \, a \)
Therefore, the correct answer is:
**sin 38° 22 mi. = sin 52° a.**