Question

1 multiple choice 1 point which of the following would be best for finding angles using law of sines? $\frac{\sin a}{a}=\frac{\sin b}{b}=\frac{\sin c}{c}$ $\frac{a}{\sin b}=\frac{b}{\sin c}=\frac{c}{\sin a}$ $\frac{a}{\sin a}=\frac{b}{\sin b}=\frac{c}{\sin c}$ $\frac{a}{\sin c}=\frac{b}{\sin a}=\frac{c}{\sin b}$

Explanation:

Step1: Recall the Law of Sines formula

The Law of Sines is \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\), where \(a\), \(b\), \(c\) are the side - lengths of a triangle and \(A\), \(B\), \(C\) are the opposite angles respectively. When we want to find angles, we can rewrite the formula as \(\sin A=\frac{a\sin B}{b}\) (by cross - multiplying \(\frac{a}{\sin A}=\frac{b}{\sin B}\)). The formula \(\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}\) is also a valid form of the Law of Sines. If we want to solve for an angle, say \(A\), we can use \(A = \sin^{-1}(\frac{a\sin B}{b})\) (from \(\frac{a}{\sin A}=\frac{b}{\sin B}\) or \(\frac{\sin A}{a}=\frac{\sin B}{b}\)).

Step2: Analyze each option

• Option1: \(\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}\) is a correct form of the Law of Sines. If we know the side - lengths \(a\), \(b\), \(c\) and one angle (say \(B\)), we can find other angles. For example, \(\sin A=\frac{a\sin B}{b}\).
• Option2: \(\frac{a}{\sin B}=\frac{b}{\sin C}=\frac{c}{\sin A}\) is an incorrect form. The correct ratio should have the side and its opposite angle.
• Option3: \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\) is the standard form of the Law of Sines. When we want to find an angle, we can rewrite it as \(\sin A=\frac{a\sin B}{b}\) (if we know \(a\), \(b\), and \(B\)).
• Option4: \(\frac{a}{\sin C}=\frac{b}{\sin A}=\frac{c}{\sin B}\) is an incorrect form. The side should be opposite to the angle in the ratio.

Answer:

\(\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}\) (Option1)