In questions 16-23 solve \(\Delta \mathrm{XYZ}\) given \(Z=46^{\circ}, \mathrm{YZ}=30 \mathrm{~m}\) and \(\mathrm{XY}=25 \mathrm{~m}\).

When solving triangles, particularly ones that are not right-angled, the Law of Sines is an invaluable tool. It expresses a relationship between the lengths of the sides of a triangle and the sines of its angles. According to the Law of Sines,
\[\begin{equation}\frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c}\end{equation}\]where A, B, and C are the angles of the triangle, and a, b, and c are the lengths of the sides opposite these angles respectively.
In our exercise, we used this principle to calculate one of the unknown angles, angle X. After identifying the known side lengths and angle, we set up a proportion using the law that involved the sides opposite the known and unknown angles. Solving for the sine of angle X, we gained a value that would help us find the actual measure of the angle with the aid of inverse trigonometric functions.

It's critical to ensure that the angle-side pairs are opposite to each other in this law, as mixing them up can lead to incorrect results. Furthermore, this law comes in very handy in the case of 'SSA' or 'AAS' triangle problems, where two sides and a non-included angle or two angles and a non-included side are given.

The triangle angle sum property is a fundamental concept in geometry. It states that the sum of the angles inside any triangle always adds up to \[\begin{equation}180^\text{ } ^\text{\circ}\end{equation}\].This property held the key to finding the third angle in our problem, angle Y. Once we calculated angle X using the Law of Sines, we determined angle Y by subtracting the measures of angles X and Z from \[\begin{equation}180^\text{ } ^\text{\circ}\end{equation}\] thus revealing the measures of all three angles. This property is useful for all triangles, regardless of their type—acute, obtuse, or right-angled—and is often used in concert with other properties or theorems, like the Law of Sines, to solve for unknown components of the triangle.

Inverse sine, also known as sinusoidal or arcsine, is a function that reverses the operation of the sine function. It takes a ratio (typically of a side length to the hypotenuse in a right-angled triangle) and returns an angle measure as the output. In our exercise, after finding the sine of angle X, we sought the actual angle. This is where the inverse sine comes into the equation. Symbolized as \[\begin{equation}\sin^{-1}\end{equation}\], it's used to determine the angle whose sine is the ratio we have calculated.
\[\begin{equation}X = \sin^{-1}(0.684) \approx 43.23^\text{ } ^\text{\circ}\end{equation}\]
This function is essential for solving triangles when a side and an angle opposite to it are known, but the angle itself is unknown. However, care must be taken with the range of values for which the inverse sine is defined, as it returns values in the range from \[\begin{equation}-90^\text{ } ^\text{\circ}\end{equation}\] to \[\begin{equation}90^\text{ } ^\text{\circ}\end{equation}\] A calculator or computer algebra system that correctly handles this function is vital for accurately solving exercises like the one in our example.