Question
n 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)Responsessin38°22 mi.=sin52°athe fraction with numerator sine 38 degrees and denominator 22 mi. is equal to the fraction with numerator sine 52 degrees and denominator Asin90°22 mi.=sin52°athe fraction with numerator sine 90 degrees and denominator 22 mi. is equal to the fraction with numerator sine 52 degrees and denominator Asin52°22 mi.=sin38°athe fraction with numerator sine 52 degrees and denominator 22 mi. is equal to the fraction with numerator sine 38 degrees and denominator Asin38°22 mi.=sin90°athe fraction with numerator sine 38 degrees and denominator 22 mi. is equal to the fraction with numerator sine 90 degrees and denominator A
Answers
GPT-4o mini
To use the Law of Sines, we have to relate the sides of the triangle to their respective opposite angles. The Law of Sines states:
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]
Given the triangle in question, we have:
- \( m\angle C = 38^\circ \)
- \( c = 22 \, \text{mi} \)
Since you want to find side \( a \) and we need angle \( A \) to use the Law of Sines, we can set up the equation using the angles and the known values.
From what we can infer about \( A \) and \( B \), we can use \( C \) and the fact that the angles in a triangle sum to \( 180^\circ \):
\[
A + B + C = 180^\circ
\]
\[
A + B + 38^\circ = 180^\circ
\]
\[
A + B = 142^\circ
\]
If we suppose \( m\angle B = 52^\circ \), then:
\[
A = 142^\circ - 52^\circ = 90^\circ
\]
Now we can correctly set up the equation to find \( a \), using the known values:
\[
\frac{a}{\sin A} = \frac{c}{\sin C}
\]
Substituting the known values, \( c = 22 \, \text{mi} \), \( C = 38^\circ \), and \( A = 90^\circ \):
\[
\frac{a}{\sin 90^\circ} = \frac{22 \, \text{mi}}{\sin 38^\circ}
\]
Since \( \sin 90^\circ = 1 \), this simplifies to:
\[
a = \frac{22 \, \text{mi}}{\sin 38^\circ}
\]
So, the equation relevant to the Law of Sines in this context would be:
\[
\sin 90^\circ / 22 \, \text{mi} = \sin 38^\circ / a
\]
This means the correct statement that aligns with this derived equation among the choices provided would be:
\[
\sin 38^\circ/22 \, \text{mi} = \sin 52^\circ/a
\]
If we break it down into the message provided, this seems to point to:
\[
\frac{\sin 38^\circ}{22 \, \text{mi}} = \frac{\sin 52^\circ}{a}
\]
So, the best option from the choices should imply this relationship.
Therefore, as per your statements, the equation correctly relating sides and angles using the Law of Sines would most likely be this form:
**Answer:**
\(\frac{\sin 38^\circ}{22 \, \text{mi}} = \frac{\sin 52^\circ}{a}\)
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]
Given the triangle in question, we have:
- \( m\angle C = 38^\circ \)
- \( c = 22 \, \text{mi} \)
Since you want to find side \( a \) and we need angle \( A \) to use the Law of Sines, we can set up the equation using the angles and the known values.
From what we can infer about \( A \) and \( B \), we can use \( C \) and the fact that the angles in a triangle sum to \( 180^\circ \):
\[
A + B + C = 180^\circ
\]
\[
A + B + 38^\circ = 180^\circ
\]
\[
A + B = 142^\circ
\]
If we suppose \( m\angle B = 52^\circ \), then:
\[
A = 142^\circ - 52^\circ = 90^\circ
\]
Now we can correctly set up the equation to find \( a \), using the known values:
\[
\frac{a}{\sin A} = \frac{c}{\sin C}
\]
Substituting the known values, \( c = 22 \, \text{mi} \), \( C = 38^\circ \), and \( A = 90^\circ \):
\[
\frac{a}{\sin 90^\circ} = \frac{22 \, \text{mi}}{\sin 38^\circ}
\]
Since \( \sin 90^\circ = 1 \), this simplifies to:
\[
a = \frac{22 \, \text{mi}}{\sin 38^\circ}
\]
So, the equation relevant to the Law of Sines in this context would be:
\[
\sin 90^\circ / 22 \, \text{mi} = \sin 38^\circ / a
\]
This means the correct statement that aligns with this derived equation among the choices provided would be:
\[
\sin 38^\circ/22 \, \text{mi} = \sin 52^\circ/a
\]
If we break it down into the message provided, this seems to point to:
\[
\frac{\sin 38^\circ}{22 \, \text{mi}} = \frac{\sin 52^\circ}{a}
\]
So, the best option from the choices should imply this relationship.
Therefore, as per your statements, the equation correctly relating sides and angles using the Law of Sines would most likely be this form:
**Answer:**
\(\frac{\sin 38^\circ}{22 \, \text{mi}} = \frac{\sin 52^\circ}{a}\)