91Ó°ÊÓ

Solving a triangle involves finding all unknown sides and angles of the triangle using given information. In this particular problem, we were given two sides and one angle, known commonly as the SAS (Side-Angle-Side) scenario. Using this information, the goal is to determine all the missing measurements. Here’s how it works in general:

You begin with the given values:

• Sides: \(a = 16.5\) and \(b = 18.2\)
• Angle: \(C = 47^{\circ}\)

By employing the Law of Cosines, you find the third side. After determining all sides, the next step is to use the Law of Sines to find the remaining angles. Understanding this step-by-step approach ensures you grasp each stage of triangle solving fully. After following the process, all side lengths and angles are known, making your triangle fully "solved."

The Law of Sines is a fundamental tool in trigonometry, especially useful in finding unknown angles or sides in a triangle when certain other measures are known. In essence, it states that the ratio of the length of a side to the sine of its opposite angle is the same for all three sides and angles in a triangle. The formula is:\[ \frac{\sin{A}}{a} = \frac{\sin{B}}{b} = \frac{\sin{C}}{c} \]
In our example, after determining the third side using the Law of Cosines, we then applied the Law of Sines to find each of the missing angles. First, it helped us calculate angle \(A\) using:\[ A = \arcsin\left(\frac{a \times \sin{C}}{c}\right) \]After determining \(A\), the same principle allows us to solve for angle \(B\), knowing each step ensures accuracy and clarity in finding the triangle's angles.

Understanding the properties of triangle angles is crucial in solving any triangle. A critical feature is that the sum of a triangle's interior angles is always \(180^{\circ}\). This rule is immensely helpful when calculating the final angle, once two angles are known. For instance, in this exercise:

• We found angle \(A\) using the Law of Sines.
• The known angle \(C\) was already provided.

Therefore, finding angle \(B\) becomes a simple subtraction problem using the formula: \[ B = 180^{\circ} - A - C \]This calculation ensures all angles correctly add up to \(180^{\circ}\), validating your results as a correctly solved triangle. Keeping this in mind simplifies the approach to every triangle problem you encounter.