Problem 21 - Precalculus Mathematics for Calculus

Question

Use the Law of sines to solve for all possible triangles that satisfy the given conditions. $$a=20, \quad c=45, \quad \angle A=125^{\circ}$$

Step-by-Step Solution

Answer

No possible triangle satisfies the given conditions because \( \sin C > 1 \).

Step 1: Identify known values and unknowns

We are given the side lengths \( a = 20 \), \( c = 45 \), and the angle \( \angle A = 125^{\circ} \). We need to find the values of \( \angle C \), \( \angle B \), and the side \( b \).

Step 2: Calculate angle C using the Law of Sines

Using the Law of Sines, \( \frac{a}{\sin A} = \frac{c}{\sin C} \). Substitute the known values to get: \[ \frac{20}{\sin 125^{\circ}} = \frac{45}{\sin C} \]. Solve for \( \sin C \): \[ \sin C = \frac{45 \cdot \sin 125^{\circ}}{20} \approx 1.4403 \]. Since the sine of an angle cannot be greater than 1, it's clear there's no possible triangle solution with these values.

Key Concepts

Triangle Solutions

Triangles can be solved using different laws and rules, depending on what information we already have. In this case, the Law of Sines is invaluable. It helps us find unknown sides and angles based on the information given in a triangle. The problem provided has side lengths and an angle, which makes it suitable for the Law of Sines. Since triangles are closed shapes with three sides and three angles, it's essential to remember that the total sum of all angles in any triangle is always 180 degrees.

When approaching triangle solutions, consider:

• What you already know (side lengths, angles).
• What you need to find (missing angles, missing side lengths).
• Which relationships or laws will help solve the problem.

It's equally important to determine if a solution is possible. The exercise exemplifies this by calculating values that must realistically fit within the properties of triangles. If calculations give unrealistic results, such as a sine value over 1, you can conclude that a triangle can't exist under the given conditions.

Angle Calculation

Calculating angles in a triangle can often be straightforward once you know certain rules and relationships. The Law of Sines is particularly helpful here, especially when you have information like an angle and its opposite side, alongside one more side or angle of the triangle.

Using the Law of Sines involves setting up ratios between the known side lengths and their opposite angles:

• The formula is: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)
• This means the ratio of a side to the sine of its opposite angle is constant.

From Step 2 of the solution, we attempted to find \( \angle C \) using the formula \( \frac{a}{\sin A} = \frac{c}{\sin C} \). Substituting known values helps simplify calculations. However, if you arrive at a sine value greater than 1, then it's impossible for that angle to exist, indicating no valid solution.

Precalculus

Precalculus forms the foundation for understanding advanced mathematical concepts, often building on algebra, geometry, and trigonometry. In this context, the Law of Sines is a crucial part of trigonometry studied in precalculus, helping to solve triangles when particular measurements are given.

Trigonometry, the study of triangles, involves several key concepts:

• Sine, Cosine, and Tangent - fundamental trigonometric ratios.
• The Pythagorean Theorem for right triangles.
• Trigonometric identities that relate different trigonometric functions.

Precalculus students should practice using these tools to solve triangle problems, keeping in mind triangle properties and trigonometric function limits. Developing a strong grasp of these foundational skills is essential for success in calculus and beyond. Solving problems like the given exercise helps reinforce understanding and application of such concepts.