Problem 26 - Precalculus : Building Concepts and Connections

Question

If at least one triangle with the given characteristics exists, solve the triangle; otherwise, state that there is no solution. If there are treo solutions, give both of them. The standard notation for labeling of triangles is used. $$A=50^{\circ}, a=10, b=8$$

Step-by-Step Solution

Answer

Therefore, using the given measures of the sides and angles, and the Law of Sines, it is possible to calculate the potential measures for the triangle. Remember to check for two possible solutions.

Step 1: Validation

Confirm if it's possible to create a triangle using the provided measures. According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This theorem doesn't apply directly here as we have an angle and two sides, not three sides.

Step 2: Calculation

Apply the Law of Sines to calculate the measure of angle B. The formula is \( \frac{a}{\sin{A}} = \frac{b}{\sin{B}} \) So, \( \sin{B} = \frac{b \cdot \sin{A}}{a} \) . Substituting the given values, we calculate \( B = \sin^{-1}(\frac{8 \cdot \sin{50}}{10}) \).

Step 3: Check for Second Solution

Since the sine of an angle is the same as the sine of its supplement, we must check for a second possible solution. If \( B \) is less than 180 degrees, then \( 180 - B \) is a second possible solution.

Step 4: Calculating the third angle

If valid, calculate the third angle C by using the formula \( C = 180 - A - B \)

Step 5: Calculation of side c

Use the Law of Sines again to find side c by cross multiplying \( \frac{c}{\sin{C}} = \frac{a}{\sin{A}} \), which gives \( c = \frac{a \cdot \sin{C}}{\sin{A}} \)

Key Concepts

Law of Sines

The Law of Sines is a fundamental principle in trigonometry for solving triangles. It provides a relationship between the lengths of the sides of a triangle and the sines of its opposite angles.
This law is crucial when dealing with non-right triangles, as it allows us to find unknown angles or sides. To apply the Law of Sines, you can use the formula:

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

Where \( a, b, \) and \( c \) are the lengths of the sides, and \( A, B, \) and \( C \) are the measures of the respective opposite angles.
This law is particularly useful for scenarios where we know:

• Two angles and one side (AAS or ASA).
• Two sides and a non-included angle (SSA or ASS, as highlighted in this exercise).

By using this formula, like in the original problem, we can calculate unknown side lengths or angle measurements, leading us to the complete solution of a triangle.

Triangle Inequality Theorem

The Triangle Inequality Theorem is an essential rule in geometry that helps determine the possibility of forming a triangle with three given sides. However, when given an angle and two sides, as in this particular problem, you might wonder how the theorem fits in. While it doesn't directly apply in an angle-side-side scenario, awareness of this theorem is critical because it gives us insight into the fundamental nature of triangles - that is, the sum of the lengths of any two sides must be greater than the length of the third side. In an ASS situation, the triangle inequality is preemptively satisfied since we attempt to solve using angles and sides. But remember:

• If any potential side length calculated doesn't result in a valid triangle (due to this theorem), it flags an issue with the data or expectations.

Thus, even in exercises where angles dominate the data, the principles embodied in the triangle inequality are part of verifying the real-world applicability of calculated shapes.

Angle-Side-Side (ASS) Problem

The angle-side-side (ASS) problem is also known as the ambiguous case problem when applying the Law of Sines. This scenario occurs when two sides and a non-included angle are known.Because of the nature of trigonometric functions, particularly sine, having two sides and an angle opposite one of the sides leads to potential ambiguity.Two possible scenarios arise from the ambiguity:

• One triangle could exist if the angle and side data permit a single geometric realization.
• Two triangles could form because of the sine's symmetry properties, reflecting across the 180-degree line.

Working through these possibilities demands thorough verification, often by checking both \( B \) and \( 180^{\circ} - B \), as this could potentially offer two valid triangles or negate possibilities based on logical impossibilities.

Ambiguous Case

The Ambiguous Case refers to the potential confusion arising from an angle-side-side configuration (ASS) in solving triangles. This ambiguity occurs primarily because the sine function in trigonometry is non-injective, meaning it maps different angles to the same value.When solving a triangle using the Law of Sines under this configuration, you might find:

• A surprising duplicate solution, where both angle \( B \) and its supplementary angle \( 180^{\circ} - B \) may satisfy your triangle's properties.
• Only one solution might legitimately work, aligning directly with your problem setup.
• No valid triangle if neither calculated potential angle satisfies logical constraints, like having a positive sum of internal angles or fitting within the triangle inequality principle.

To address the ambiguous case, we validate all potential solutions and eliminate infeasible triangles, ensuring any used solutions adhere to mathematical and real-world logic.