91Ó°ÊÓ

The Law of Sines is a helpful tool in solving triangles, especially when dealing with non-right triangles. This law states that in any triangle, the ratios of each side's length to the sine of its opposite angle are equivalent. Mathematically, this can be represented as:\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]In the context of the problem, we used the Law of Sines to find angle \(B\). By arranging the formula and substituting known values for sides \(a\), \(b\), and angle \(A\), we can solve for \(\sin B\) and subsequently \(B\) itself. This approach allows us to explore multiple potential triangle configurations by considering different angle values.

Converting angle measurements from degrees and minutes to decimal form is essential in trigonometric calculations to maintain simplicity and consistency. Here's a quick snapshot of how it works:- Degrees are typically measured in a smaller unit called minutes, where 1 degree equals 60 minutes.- To convert minutes into degrees, divide the minutes by 60. For example, in our exercise, angle \(A\) was given as \(38^{\circ} 40'\). We convert 40 minutes to degrees by performing: \[40' = \frac{40}{60} = 0.6667^{\circ}\]Therefore, angle \(A\) becomes \(38.6667^{\circ}\) in decimal form. This simplification is crucial when using trigonometric functions, such as sine and cosine, which expect angle input in decimal degrees.

Solving for angle measures often involves using inverse trigonometric functions, such as the inverse sine function \(\sin^{-1}\). These functions can help you find the angle when you know its sine value.In our solution, we found \(\sin B \approx 0.7606\). To determine angle \(B\), we use:\[B = \sin^{-1}(0.7606)\]By calculating, we find \(B\) equals \(49.52^{\circ}\). It's important to remember that each sine value can correspond to two possible angles in different triangles:

• The angle \(B\) itself
• The supplementary angle \(180^{\circ} - B\)

Thus, calculations should check both potential scenarios, ensuring they make logical sense within the triangle.

One fundamental rule in geometry is the triangle angle sum property. This property states that the sum of all interior angles in a triangle is always \(180^{\circ}\). Understanding this property helps us validate potential solutions when solving triangles.In our case, for each possible angle \(B\) derived from the inverse sine calculation, we ensured:\[A + B + C = 180^{\circ}\]For example:

• If \(B = 49.52^{\circ}\), then \(A + B = 88.1867^{\circ}\). Thus, \(C = 180^{\circ} - 88.1867^{\circ} = 91.8133^{\circ}\).
• If \(B = 130.48^{\circ}\), then \(A + B = 169.1467^{\circ}\). Thus, \(C = 180^{\circ} - 169.1467^{\circ} = 10.8533^{\circ}\).

Using this property ensures that each possible set of angles forms a valid triangle, making it a critical check in solving such problems.