91Ó°ÊÓ

The Law of Sines helps us to determine unknown angles or sides in a triangle. This law connects the ratios of the lengths of the sides to the sines of their opposite angles in a triangle. The formula is expressed as \[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}. \]
In our given problem, we used the Law of Sines to find angle B: \[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)}. \]
Plugging in the values, we get: \[ \frac{15.6}{\sin(89^{\circ})} = \frac{18.4}{\sin(B)}. \]
Simplifying further, \[ \frac{15.6}{1} = \frac{18.4}{\sin(B)}. \]
This equation reveals a problem when trying to solve for \sin(B). We find \[ \sin(B) = \frac{18.4}{15.6} = 1.1795. \]
Since \sin(x) cannot be greater than 1, it suggests that there is no possible triangle with the given side lengths and angle. This is a crucial concept when applying the Law of Sines, as it can also help to understand when a triangle cannot exist.

The Triangle Inequality Theorem is fundamental to determine whether a set of sides can form a triangle. It states that the sum of any two sides of a triangle must be greater than the third side. This theorem applies to all triangles. Here’s how you can check it:

• \(a + b > c\)
• \(a + c > b\)
• \(b + c > a\)

In our exercise, we use sides \(a = 15.6\) inches and \(b = 18.4\) inches. However, we do not yet know the third side, \(c\). Therefore, we proceed with the Law of Sines to determine the angles first.
During our calculation, we see that obtaining \(\sin(B) > 1\) means that such a triangle is not possible. This result aligns with the Triangle Inequality Theorem indirectly, as it hints at the impossibility of such side lengths forming a triangle.

Angle calculation in a triangle can be done using different methods such as the Law of Sines, the Law of Cosines, or by knowing that the sum of interior angles in a triangle is always \(180^{\circ}\).
For example, using the Law of Sines, we tried to find \(B\) with the formula: \[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)}. \]
By solving this equation, we found \[ \sin(B) = \frac{18.4}{15.6} = 1.1795. \]
Since the sine of an angle cannot exceed 1, this indicated an error or impossibility. These calculations are crucial as they verify whether a triangle can exist with the given dimensions.