Problem 51 - A Graphical Approach to Precalculus with Limits

Question

Solve triangle. There may be two, one, or no such triangle. $$A=38^{\circ} 40^{\prime}, a=9.72 \text { kilometers, } b=11.8 \text { kilometers }$$

Step-by-Step Solution

Answer

Two possible triangles exist; B can be 49.52° or 130.48°.

Step 1: Convert Angles to Decimal Degrees

Convert the given angle for angle \(A\) from degrees and minutes to decimal form.We have \(A = 38^{\circ} 40'\).To convert minutes to degrees, divide by 60: \(40' = \frac{40}{60} = 0.6667^{\circ}\).So, \(A = 38.6667^{\circ}\).

Step 2: Use the Law of Sines to Find Angle B

Use the Law of Sines: \(\frac{a}{\sin A} = \frac{b}{\sin B}\).Plug in the known values: \(\frac{9.72}{\sin 38.6667^{\circ}} = \frac{11.8}{\sin B}\).First, calculate \(\sin 38.6667^{\circ} \approx 0.6257\).Then solve for \(\sin B\): \(\sin B = \frac{11.8 \times 0.6257}{9.72} \approx 0.7606\).

Step 3: Solve for Angle B

Using \(\sin B \approx 0.7606\), calculate \(B\) using the inverse sine function:\(B = \sin^{-1}(0.7606) \approx 49.52^{\circ}\).This gives us two possible readings: \(49.52^{\circ}\) or \(180^{\circ} - 49.52^{\circ} = 130.48^{\circ}\).

Step 4: Check the Sum of Angles for Validity

Check these angles with the law of triangles, \(A + B < 180^{\circ}\).1. For \(B = 49.52^{\circ}\), \(A + B = 38.6667^{\circ} + 49.52^{\circ} = 88.1867^{\circ}\) which is valid.2. For \(B = 130.48^{\circ}\), \(A + B = 38.6667^{\circ} + 130.48^{\circ} = 169.1467^{\circ}\) which is also valid.

Step 5: Calculate Possible Angles C

For each valid value of \(B\) calculate \(C = 180^{\circ} - A - B\).1. If \(B = 49.52^{\circ}\), then \(C = 180^{\circ} - 38.6667^{\circ} - 49.52^{\circ} = 91.8133^{\circ}\).2. If \(B = 130.48^{\circ}\), then \(C = 180^{\circ} - 38.6667^{\circ} - 130.48^{\circ} = 10.8533^{\circ}\).

Step 6: Use Law of Sines to Calculate Side c

For each scenario, calculate side \(c\) using \(\frac{c}{\sin C} = \frac{a}{\sin A}\).1. If \(C = 91.8133^{\circ}\), solve \(c = \frac{9.72 \times \sin 91.8133^{\circ}}{\sin 38.6667^{\circ}} \approx 15.55\) kilometers.2. If \(C = 10.8533^{\circ}\), solve \(c = \frac{9.72 \times \sin 10.8533^{\circ}}{\sin 38.6667^{\circ}} \approx 2.86\) kilometers.

Key Concepts

Law of Sines

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.

Angle Conversion

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.

Inverse Sine Function

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.

Triangle Angle Sum Property

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.