Question

law of sines: \\(\frac{\sin(a)}{a} = \frac{\sin(b)}{b} = \frac{\sin(c)}{c}\\)
triangle qrs: side qr is 2.4, side qs is 3.5, angle at r is 100°
which equation is true for triangle qrs?

1. \\(\frac{\sin(100°)}{3.5} = \frac{\sin(s)}{2.4}\\)
2. \\(\frac{\sin(100°)}{3.5} = \frac{\sin(q)}{2.4}\\)
3. \\(\frac{\sin(100°)}{2.4} = \frac{\sin(s)}{3.5}\\)
4. \\(\frac{\sin(100°)}{2.4} = \frac{\sin(q)}{3.5}\\)

Explanation:

Step1: Recall the Law of Sines formula

The Law of Sines states that in a triangle \( \triangle ABC\), \(\frac{\sin(A)}{a}=\frac{\sin(B)}{b}=\frac{\sin(C)}{c}\), where \(A\), \(B\), \(C\) are angles and \(a\), \(b\), \(c\) are the lengths of the sides opposite to these angles respectively.
In \(\triangle QRS\), let's identify the sides and angles. The side opposite to \(\angle R = 100^{\circ}\) is \(QS = 3.5\). The side opposite to \(\angle S\) is \(QR=2.4\).

Step2: Apply the Law of Sines

According to the Law of Sines, \(\frac{\sin(R)}{QS}=\frac{\sin(S)}{QR}\). Substituting \(R = 100^{\circ}\), \(QS = 3.5\) and \(QR = 2.4\) into the formula, we get \(\frac{\sin(100^{\circ})}{3.5}=\frac{\sin(S)}{2.4}\)

Answer:

\(\frac{\sin(100^{\circ})}{3.5}=\frac{\sin(S)}{2.4}\) (the first option)