Question

1. choose the law and the formula that would be used to solve the triangle.

law
formula
a. law of cosines
b. \\(\frac{\sin a}{a} = \frac{\sin b}{b}\\)
c. \\(\frac{\sin a}{a} = \frac{\sin c}{c}\\)
d. \\(\frac{\sin b}{b} = \frac{\sin c}{c}\\)
e. law of sines

Explanation:

Step1: Identify the given information

We have a triangle with angle \( B = 84^\circ \), angle \( C = 44^\circ \), and side \( c = 4 \) cm (length of \( AB \)). We need to solve the triangle, so we use the Law of Sines or Law of Cosines. The Law of Sines relates the ratios of sides and their opposite angles (\( \frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c} \)), and the Law of Cosines is for when we have two sides and the included angle or three sides. Here, we have two angles and a side, so Law of Sines (option e) is appropriate.

Step2: Match the formula with the Law of Sines

Looking at the options, the Law of Sines formula that uses angle \( B \), angle \( C \), side \( b \), and side \( c \) is \( \frac{\sin B}{b}=\frac{\sin C}{c} \) (option d), since angle \( B \) is opposite side \( b \), angle \( C \) is opposite side \( c \).

Answer:

Law: e. Law of Sines
Formula: d. \(\boldsymbol{\frac{\sin B}{b}=\frac{\sin C}{c}}\)