Question

which represents the value of c? law of sines: $\frac{sin(a)}{a}=\frac{sin(b)}{b}=\frac{sin(c)}{c}$

Explanation:

Step1: Identify angles and side in the law of sines

In the triangle, let \(a = 10\) cm, \(A = 45^{\circ}\), \(B=40^{\circ}\), and we want to find side \(c\). According to the law of sines \(\frac{\sin(A)}{a}=\frac{\sin(B)}{b}=\frac{\sin(C)}{c}\). First, we know that the sum of angles in a triangle is \(180^{\circ}\), so \(C=180^{\circ}-45^{\circ}- 40^{\circ}=95^{\circ}\). But we can also use the ratio \(\frac{\sin(A)}{a}=\frac{\sin(B)}{c}\) (re - arranging for \(c\)).

Step2: Rearrange the law - of - sines formula for \(c\)

From \(\frac{\sin(A)}{a}=\frac{\sin(B)}{c}\), we can cross - multiply to get \(c\times\sin(A)=a\times\sin(B)\), then \(c = \frac{a\sin(B)}{\sin(A)}\). Substituting \(a = 10\) (assuming the number 10 in the problem is the length of the side), \(A = 45^{\circ}\), and \(B = 40^{\circ}\), we have \(c=\frac{10\sin(40^{\circ})}{\sin(45^{\circ})}\). If we assume the number 3 in the options is a mis - type of 10, the correct formula for \(c\) based on the law of sines is \(c=\frac{(10)\sin(40^{\circ})}{\sin(45^{\circ})}\).

Answer:

\(c=\frac{(10)\sin(40^{\circ})}{\sin(45^{\circ})}\) (assuming the number in the numerator should be 10 instead of 3 as per the problem - setup and law of sines application) and among the given options, if we assume the number 3 is a wrong number and should be the side - length value, the closest correct form is \(c=\frac{(10)\sin(40^{\circ})}{\sin(45^{\circ})}\), which is similar to the first option if we correct the number in the numerator. So if we assume the correct form with the actual side - length value, the answer is the first option with the correct number in the numerator. If we consider the numbers as they are in the options, there is an error in the problem setup as the side - length value seems misrepresented, but conceptually the form should be \(c=\frac{(side - length)\sin(40^{\circ})}{\sin(45^{\circ})}\).