Question

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

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

Explanation:

Step1: Identify the given information

We have a triangle with two angles (\( \angle A = 52^\circ \), \( \angle C = 44^\circ \)) and one side (\( c = 4 \) cm). To solve the triangle (find unknown sides and angles), we need to determine the appropriate law.

Step2: Recall the laws for solving triangles

• Law of Sines: \( \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} \) (relates angles and their opposite sides, useful when we have angle - side pairs or two angles and a side).
• Law of Cosines: \( c^{2}=a^{2}+b^{2}-2ab\cos C \) (useful when we have two sides and the included angle or three sides).

Since we have two angles (\( \angle A \), \( \angle C \)) and one side (\( c \)), the Law of Sines is appropriate.

Step3: Match the formula with the Law of Sines

The Law of Sines formula is \( \frac{\sin A}{a}=\frac{\sin C}{c} \) (or other angle - side pairs). Looking at the options, option d is \( \frac{\sin A}{a}=\frac{\sin C}{c} \) and option e is the Law of Sines.

Answer:

Law: e. Law of Sines
Formula: d. \( \frac{\sin A}{a}=\frac{\sin C}{c} \)