Problem 26 - Algebra and Trigonometry

Question

Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. $$ b=73, \quad c=82, \quad \angle B=58^{\circ} $$

Step-by-Step Solution

Answer

There is one possible triangle with sides \( a \approx 66.0 \), \( b = 73 \), \( c = 82 \) and angles \( A = 50^{\circ} \), \( B = 58^{\circ} \), \( C = 72^{\circ} \).

Step 1: Write the Law of Sines Formula

The Law of Sines relates the sides of a triangle to the sines of its angles.The formula is \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \).

Step 2: Calculate \( \sin C \)

Using the formula: \( \frac{b}{\sin B} = \frac{c}{\sin C} \). Substitute \( b = 73 \), \( c = 82 \), and \( \angle B = 58^{\circ} \).Therefore, \( \frac{73}{\sin 58^{\circ}} = \frac{82}{\sin C} \).Solving for \( \sin C \), we get \( \sin C = \frac{82 \cdot \sin 58^{\circ}}{73} \).

Step 3: Solve for \( C \)

Calculate \( \sin 58^{\circ} \), which is approximately 0.8480.Now substituting, \( \sin C = \frac{82 \cdot 0.8480}{73} = 0.9521 \).Since \( \sin C \) is positive and less than 1, we find \( C \) using \( C = \arcsin(0.9521) \), which gives \( C \approx 72^{\circ} \).

Step 4: Determine \( \angle A \)

The sum of angles in a triangle is \( 180^{\circ} \). Thus, \( \angle A = 180^{\circ} - \angle B - \angle C \).\( \angle A = 180^{\circ} - 58^{\circ} - 72^{\circ} = 50^{\circ} \).

Step 5: Solve for \( a \)

Using the Law of Sines: \( \frac{a}{\sin A} = \frac{b}{\sin B} \).\( \frac{a}{\sin 50^{\circ}} = \frac{73}{\sin 58^{\circ}} \).\( a = \frac{73 \cdot \sin 50^{\circ}}{0.8480} \).Calculate \( \sin 50^{\circ} \), which is approximately 0.7660. Substituting gives:\( a \approx \frac{73 \cdot 0.7660}{0.8480} = 66.0 \).

Step 6: Check for Possible Ambiguity

Since \( C \) was obtained using an \( \arcsin \), check for a possible second solution: \( C' = 180^{\circ} - 72^{\circ} = 108^{\circ} \).If \( C' = 108^{\circ} \), then \( \angle A' + \angle B + \angle C' \) would exceed \( 180^{\circ} \).Thus, no second triangle is possible.

Key Concepts

Triangle Solving with the Law of Sines

In trigonometry, solving triangles involves finding unknown side lengths and angles. The Law of Sines is a powerful tool for this task, especially in oblique triangles (non-right triangles). It provides a relationship between the length of sides of a triangle and the sines of its angles. This law is expressed as:\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]Where:

• \( a, b, c \) are the side lengths
• \( A, B, C \) are the opposite angles

To apply this concept, we use known side and angle measurements to find the unknowns. It helps us figure out potential configurations of a triangle, giving solutions that satisfy given conditions.

Angle Calculation Using the Law of Sines

Angle calculation is crucial when using the Law of Sines. In our exercise, we began with given values for side lengths and an angle (\( b=73,\ c=82,\ \angle B=58^\circ \)). Our task was to determine the other two angles of the triangle. First, we rearranged the formula to solve for one of the unknown angles, \( C \):\[ \frac{b}{\sin B} = \frac{c}{\sin C} \]By substituting known values and calculating \( \sin C \), we could find \( C \) by taking the arcsin of the result. We found that \( C \approx 72^\circ \). Finally, knowing that the sum of all angles in a triangle is \( 180^\circ \), we easily calculated \( \angle A \) by adding \( \angle B \) and \( \angle C \) and subtracting this total from \( 180^\circ \). This gave us \( \angle A \approx 50^\circ \).

Understanding the Ambiguous Case of the Law of Sines

The ambiguous case occurs under specific conditions in triangle solving using the Law of Sines. It arises when two sides and a non-included angle (SAS) are known, potentially leading to two different triangle solutions.In our exercise, after calculating \( C \), we checked for ambiguity by considering \( C' = 180^\circ - C \). If \( C' \) provides another valid angle, this could indicate the presence of a second triangle. However, we found that if \( C' = 108^\circ \), then the total of angles (\( \angle A' + \angle B + \angle C' \)) would exceed \( 180^\circ \). Thus, no secondary solution was possible for this scenario. This means the exercise does not present an ambiguous case.

Exploring Trigonometry in Triangles

Trigonometry is the study of relationships between the angles and sides of triangles. It provides critical tools like the Law of Sines and the Law of Cosines, essential for solving triangles. By utilizing trigonometric ratios—in this case, sines—we can solve complex problems and calculate unknown triangle measurements efficiently and accurately. Applying trigonometry in this example involved using angles and given side lengths to explore possible triangle dimensions. The correct use of trigonometric principles ensured that we identified all possible scenarios while adhering to the geometric rules of triangles. This systematic approach underscores the significance of trigonometry in mathematical problem-solving.