Question

use the triangle shown on the right to answer the question. according to the law of sines, which of the following statement(s) are true? check all that apply.
( b sin ( a ) = a sin ( b ) )
( c sin ( c ) = b sin ( b ) )
( \frac { a } { sin ( a ) } = \frac { b } { sin ( b ) } )
( \frac { b } { sin ( b ) } = \frac { c } { sin ( c ) } )

Explanation:

Step1: Recall the Law of Sines formula

The Law of Sines states that in a triangle \(\triangle ABC\) with sides \(a\), \(b\), \(c\) opposite to angles \(A\), \(B\), \(C\) respectively, \(\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}\).

Step2: Analyze \(b\sin(A)=a\sin(B)\)

From \(\frac{a}{\sin(A)}=\frac{b}{\sin(B)}\), cross - multiply. We get \(b\sin(A)=a\sin(B)\) (by cross - multiplying \(a\times\sin(B)=b\times\sin(A)\)).

Step3: Analyze \(c\sin(C)=b\sin(B)\)

From \(\frac{b}{\sin(B)}=\frac{c}{\sin(C)}\), cross - multiply to get \(b\sin(C)=c\sin(B)\), not \(c\sin(C)=b\sin(B)\).

Step4: Analyze \(\frac{a}{\sin(A)}=\frac{b}{\sin(B)}\)

This is directly from the Law of Sines formula \(\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}\).

Step5: Analyze \(\frac{b}{\sin(B)}=\frac{c}{\sin(C)}\)

This is also directly from the Law of Sines formula \(\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}\).

Answer:

\(b\sin(A)=a\sin(B)\), \(\frac{a}{\sin(A)}=\frac{b}{\sin(B)}\), \(\frac{b}{\sin(B)}=\frac{c}{\sin(C)}\)