Question

question 15 of 26 which of the following equations correctly represents the law of sines? a. ( a = \frac{a sin b}{sin b} ) b. ( a = \frac{b sin b}{sin a} ) c. ( a = \frac{b sin a}{sin b} ) d. ( a = \frac{a sin a}{sin b} )

Explanation:

Step1: Recall the Law of Sines

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

Step2: Derive the expression for \(a\)

From \(\frac{a}{\sin A}=\frac{b}{\sin B}\), we can cross - multiply to solve for \(a\). Cross - multiplying gives \(a\times\sin B = b\times\sin A\). Then, dividing both sides by \(\sin B\) (assuming \(\sin B
eq0\)), we get \(a=\frac{b\sin A}{\sin B}\).

Step3: Analyze the options

• Option A: \(a = \frac{a\sin B}{\sin B}\) simplifies to \(a = a\), which is a tautology but not the Law of Sines representation for finding \(a\).
• Option B: \(a=\frac{b\sin B}{\sin A}\) does not match the derived formula from the Law of Sines.
• Option C: \(a=\frac{b\sin A}{\sin B}\) matches the formula we derived from the Law of Sines.
• Option D: \(a=\frac{a\sin A}{\sin B}\) does not match the Law of Sines formula.

Answer:

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