Question

which equation represents the law of sines for \\( \triangle abc \\)?
\\( \frac{\sin a}{b}=\frac{\sin b}{a} \\)
\\( c\sin c = b\sin b \\)
\\( \frac{\sin a}{a}=\frac{\sin b}{b} \\)
\\( b\sin b = a\sin a \\)

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, the formula is \(\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}\).

Step2: Analyze each option

• For the option \(\frac{\sin A}{b}=\frac{\sin B}{a}\), it does not match the Law of Sines formula.
• For the option \(c\sin C = b\sin B\), from \(\frac{\sin B}{b}=\frac{\sin C}{c}\) we get \(c\sin B=b\sin C\), so this option is wrong.
• For the option \(b\sin B=a\sin A\), from \(\frac{\sin A}{a}=\frac{\sin B}{b}\) we get \(b\sin A = a\sin B\), so this option is wrong.
• The option \(\frac{\sin A}{a}=\frac{\sin B}{b}\) is in line with the Law of Sines formula \(\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}\).

Answer:

$\frac{\sin A}{a}=\frac{\sin B}{b}$