To solve the two triangles shown in the image, we must decide whether to use the Law of Sines or the Law of Cosines. Let's analyze each case:

(a) Triangle:

Given:

• $a = 12$
• $b = 16$
• $c = 18$

Here, you have all three sides of the triangle, but no angles are given. The Law of Cosines is more suitable in this situation because it allows you to find the angles when all three sides are known. Specifically, the Law of Cosines relates the sides and one angle as:

$\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}$ You can use this to find one of the angles first, and then proceed.

(b) Triangle:

Given:

• $b = 35^\circ$
• $C = 55^\circ$
• $c = 18$

In this triangle, you are given two angles and one side. This is a case where the Law of Sines would be more appropriate because the Law of Sines works best when you have two angles and one side (or two sides and a non-included angle). The Law of Sines states:

$\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$ You can use this to find the missing sides or angles.

Summary:

• Use the Law of Cosines for part (a) because all sides are given and you need to find the angles.
• Use the Law of Sines for part (b) because you have two angles and a side.

Would you like more detailed steps for solving each part?

Questions to expand on:

1. How can we find the remaining sides and angles after applying the Law of Cosines?
2. Can you explain the derivation of the Law of Cosines formula?
3. What is the significance of knowing two angles when applying the Law of Sines?
4. What steps should be taken if two sides and one non-included angle are given?
5. How does the Law of Sines handle cases where there is an "ambiguous case" (SSA)?

Tip:

When using the Law of Sines, always check whether there could be two possible solutions for the triangle, especially if given two sides and an angle (SSA case).