Problem 50 - A Graphical Approach to Precalculus with Limits

Question

Solve triangle. There may be two, one, or no such triangle. $$C=68.5^{\circ}, c=258 \text { centimeters, } b=386 \text { centimeters }$$

Step-by-Step Solution

Answer

No triangle can be formed with the given measurements.

Step 1: Understand Given Data and Setup

We are given a triangle with angle \( C = 68.5^{\circ} \), side \( c = 258 \) cm, and side \( b = 386 \) cm. We aim to find the remaining parts of the triangle: angle \( B \), angle \( A \), and side \( a \).

Step 2: Determine Triangle Type Using Law of Sines

Using the Law of Sines, we have \( \frac{c}{\sin C} = \frac{b}{\sin B} \). This can be rewritten as \( \sin B = \frac{b \cdot \sin C}{c} \). Calculate \( \sin B = \frac{386 \cdot \sin(68.5^{\circ})}{258} \).

Step 3: Calculate \( \sin(68.5^{\circ}) \)

Calculate \( \sin(68.5^{\circ}) \approx 0.9272 \). Now compute \( \sin B \approx \frac{386 \cdot 0.9272}{258} \approx 1.387 \). Since \( \sin B \) cannot be greater than 1, there is a mistake in measurement or setup.

Step 4: Conclusion

Since \( \sin B > 1 \), this indicates no triangle can be formed with the given measurements using these angle-side relationships. This confirms no such triangle exists.

Key Concepts

Triangle Solving

When solving triangles, especially in trigonometry, the goal is to determine unknown angles and sides based on the given data. In many problems, the Law of Sines or the Law of Cosines are used to find missing parts of a triangle. These laws relate the angles and sides of a triangle in explicit equations, which provide a foundation for solving unknowns.

In the given problem, the task was to solve a triangle with specific measurements of one angle and two sides. "Solving" means finding all missing angles and the side not provided. However, it's important to ensure that the given dimensions can actually form a triangle.

• Understand what is known: Identify known angles and sides.
• Apply the Law of Sines or Cosines to find missing values.
• Verify if the triangle formation is feasible based on angle relationships and side lengths.

If calculations show impossible values (like a sine ratio exceeding 1), then it may indicate that a triangle with the provided characteristics cannot exist.

Angle-Side Relationships

The relationship between angles and sides in a triangle is a fundamental concept in trigonometry. These relationships are governed by specific rules, such as the Law of Sines and the Law of Cosines. In the context of solving triangles, these laws are used to find unknown angles or sides given partial information.

The Law of Sines states: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \), which means the ratio of a side to the sine of its opposite angle is constant for all sides of a triangle. This can help predict whether the given measurements can form a valid triangle.

• If the sine of angle B calculated exceeds 1, it implies an error, as sine values cannot be greater than 1.
• Cross-checks are always essential to ensure given angles and sides can exist within a single triangle.
• Understanding these relationships helps to judge the possibility of triangle formulation.

The key to handling trigonometry problems effectively is having a strong grasp of these angle-side interactions and ensuring logical consistency in the geometric shape.

Trigonometry

Trigonometry is all about understanding the relationships between the angles and sides of triangles. It's a branch of mathematics that plays a crucial role in solving triangle problems, especially when dealing with angles and their corresponding sides.

In the particular problem at hand, trigonometry is used to determine whether a triangle can be formed with the given dimensions. To do this, we utilize key trigonometric functions and laws:

• Use of sine, which relates the measures of triangles' angles to their sides.
• The Law of Sines, a primary tool for finding missing parts of a triangle.
• Analysis of results, ensuring sine outputs are within the valid range of [-1, 1], validating results before concluding.

Any anomaly, like a sine value going out of range, typically indicates that the initial parameters do not define a real triangle. Mastery of trigonometric principles equips students to not only solve problems accurately but to also understand when given information indicates a non-viable scenario.