Question

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

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

Explanation:

Step1: Identify the given information

We have a triangle with side \( a = 4.5 \) cm, side \( b = 5.7 \) cm, and angle \( A = 52^\circ \). We need to choose the law and formula to solve the triangle.

Step2: Analyze the Law of Sines and Law of Cosines

The Law of Sines states that \( \frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c} \). The Law of Cosines is used when we have two sides and the included angle or three sides. Here, we have a side - angle - side (a, A, b) situation, so the Law of Sines is applicable.

Step3: Match the formula with the Law of Sines

Among the given formulas, the formula \( \frac{\sin A}{a}=\frac{\sin B}{b} \) (option b) is a part of the Law of Sines and is relevant here as we know \( a = 4.5 \), \( A=52^\circ \) and \( b = 5.7 \), and we can use this formula to find angle \( B \) or other unknowns. Also, the law to use is the Law of Sines (option c).

Answer:

Law: c. Law of Sines
Formula: b. \(\frac{\sin A}{a}=\frac{\sin B}{b}\)