Problem 37 - A Graphical Approach to Precalculus with Limits

Question

Determine the number of triangles ABC possible with the given parts. \(a=31, b=26, B=48^{\circ}\)

Step-by-Step Solution

Answer

One triangle is possible with angles \(A = 38.49^{\circ}\), \(B = 48^{\circ}\), and \(C = 93.51^{\circ}\).

Step 1: Identify the Known Parts

We know the lengths of sides \(a = 31\) and \(b = 26\), and the measure of angle \(B = 48^{\circ}\). We are tasked with finding the possible number of triangles given these conditions.

Step 2: Apply the Law of Sines

To determine whether a triangle is possible, apply the Law of Sines: \(\frac{a}{\sin A} = \frac{b}{\sin B}\). Since \(B = 48^{\circ}\), calculate \(\sin B\): \(\sin 48^{\circ} \approx 0.7431\). Now compute \(\sin A\): \(\sin A = \frac{b \cdot \sin B}{a} = \frac{26 \cdot 0.7431}{31} \approx 0.6233\).

Step 3: Determine Feasibility of Angle A

A triangle is possible when \(\sin A \leq 1\). Here \(\sin A \approx 0.6233\) which satisfies this condition. Thus, \(A\) is a valid angle. Since \(\sin A\) is less than 1, there are potentially two solutions: acute angle \(A\) and obtuse angle \(A' = 180^{\circ} - A\).

Step 4: Calculate Possible Angles for A

Calculate the acute angle \(A\) using \(\sin^{-1}\): \(A = \sin^{-1}(0.6233) \approx 38.49^{\circ}\). For a possible obtuse angle, \(A' = 180^{\circ} - 38.49^{\circ} = 141.51^{\circ}\).

Step 5: Determine Validity of 3rd Angle

If \(A \) or \(A'\) leads to a valid triangle, \(C = 180^{\circ} - A - B\). Calculate \(C\) for the acute solution: \(C = 180^{\circ} - 38.49^{\circ} - 48^{\circ} = 93.51^{\circ}\), which is possible. For the obtuse solution: \(C = 180^{\circ} - 141.51^{\circ} - 48^{\circ} = -9.51^{\circ}\), which is not possible.

Key Concepts

Law of Sines

The Law of Sines is a fundamental principle used in trigonometry to find unknown measurements in triangles. It relates the lengths of a triangle's sides to the sines of its angles. This is given by the formula:

• \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)

Where \( a, b, c \) are the sides of the triangle, and \( A, B, C \) are the opposite angles respectively. In our exercise, we used the Law of Sines to find the sine of angle \( A \).
This step involves using the known values, which are the lengths of sides \( a = 31 \) and \( b = 26 \), and the measure of angle \( B = 48^{\circ} \). By substituting into the Law of Sines, we calculate \( \sin A \), which helps us further understand and solve the triangle.

Angle Calculation

To determine angle \( A \), we use its sine value computed from the Law of Sines. Sine is a trigonometric function that measures the ratio of the opposite side to the hypotenuse in a right-angled triangle. In this problem, \( \sin A \approx 0.6233 \).
To find angle \( A \) itself, we apply the inverse sine function, or \( \sin^{-1} \), which gives us the angle whose sine is \( 0.6233 \). This results in \( A \approx 38.49^{\circ} \). There's an interesting twist here; every sine value corresponds to two potential angles within a full circle, an acute and an obtuse:

• Acute angle: \( A \approx 38.49^{\circ} \)
• Obtuse angle: \( A' = 180^{\circ} - 38.49^{\circ} = 141.51^{\circ} \)

This gives us the potential for different triangle configurations, which we must check for feasibility.

Triangle Feasibility

Determining whether a potential triangle is feasible involves checking if the calculated angles can form a valid triangle. According to triangle properties, the sum of the angles in any triangle must be exactly \( 180^{\circ} \).
For our calculations, we first check the acute angle setup:

• \( A \approx 38.49^{\circ} \)
• \( B = 48^{\circ} \)
• \( C = 180^{\circ} - 38.49^{\circ} - 48^{\circ} = 93.51^{\circ} \)

These angles add up correctly, indicating a valid triangle can indeed be formed.
Next, we examine the obtuse scenario:

• \( A' = 141.51^{\circ} \)
• \( B = 48^{\circ} \)
• \( C = 180^{\circ} - 141.51^{\circ} - 48^{\circ} = -9.51^{\circ} \)

Here, the sum does not accommodate a valid third angle since negative angles are not possible in geometric contexts. Hence, only the acute angle case results in a feasible triangle, confirming that one triangle is possible.