Problem 7 - Algebra and Trigonometry with Analytic Geometry

Question

Solve \(\triangle A B C\). $$\beta=150^{\circ}, \quad a=150, \quad c=30$$

Step-by-Step Solution

Answer

The triangle cannot exist with the given dimensions.

Step 1: Understand the Given Information

In triangle \( \triangle ABC \), we are given \( \beta = 150^{\circ} \), \( a = 150 \), and \( c = 30 \). These represent angle \( B \) and sides \( a = BC \) and \( c = AB \) respectively.

Step 2: Use the Law of Sines

The Law of Sines states \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). We'll start by using this to find another angle. Substitute the known values: \( \frac{150}{\sin A} = \frac{30}{\sin 150^{\circ}} \). Since \( \sin 150^{\circ} = \sin 30^{\circ} = \frac{1}{2} \), the equation becomes \( \frac{150}{\sin A} = 60 \).

Step 3: Solve for Angle A

Rearrange to find \( \sin A \): \( \sin A = \frac{150}{60} = 2.5 \). This is invalid since the sine of an angle cannot exceed 1, indicating the triangle cannot exist with the given dimensions.

Step 4: Confirmation of No Solution

The invalid sine value means there is no possible angle \( A \) that satisfies the triangle inequality under the given conditions. Thus, no such triangle can exist with the provided values.

Key Concepts

Law of Sines

The Law of Sines is an essential formula in trigonometry, especially useful for finding unknown angles or sides in any triangle. It's given by the equation \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \), where \( a \), \( b \), and \( c \) are the lengths of the sides opposite to angles \( A \), \( B \), and \( C \), respectively.
This law helps us relate the ratios of sides to the sines of their opposing angles. Particularly, it is most helpful when dealing with non-right triangles, known as oblique triangles.

• First, it allows solving triangles when we are given either two angles and one side (AAS or ASA conditions) or two sides and a non-enclosed angle (SSA condition).
• Secondly, when applying the Law of Sines, it’s crucial to understand that it only works accurately if the calculated sine value lies between 0 and 1, since sine is a trigonometric function with output constrained to this range.

For our exercise, substituting the given values resulted in an unattainable sine value greater than 1, revealing the impossibility of the triangle formation.

Triangle Inequality

The triangle inequality is a fundamental principle that governs the feasibility of triangle formation with three given side lengths. According to this principle:

• The sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.
• This principle effectively forbids the possibility of forming a triangle if this condition isn’t met.

In mathematical terms, for a triangle with sides \( a \), \( b \), and \( c \), the following must hold true:

• \( a + b > c \)
• \( a + c > b \)
• \( b + c > a \)

In the context of our exercise, examining the angle β = 150° combined with sides \( a = 150 \) and \( c = 30 \), using the sine law led us to a contradiction in the expected value. This apparent contradiction reinforces that certain side and angle combinations cannot fulfill triangle inequalities, confirming that no such triangle can exist.

Angles and Sides in Triangles

Understanding how angles and sides interact within a triangle is crucial to solving any trigonometric problem. In every triangle, the sum of its internal angles is always 180 degrees. Therefore, if two angles are known, the third can simply be found by subtracting the sum of the known angles from 180 degrees.
Moreover, when working with triangles, it’s essential to match angles and sides correctly:

• The largest angle is always opposite the longest side.
• The smallest angle lies opposite the shortest side.
• Angle magnitude determines the side length in anticipation, dictating the triangle's scalability.

In our specific example, the given data provided challenges, revealing inherent inconsistencies, such as suggesting an angle with a sine greater than 1. Such scenarios often highlight an impossible geometric configuration, indicating that either the measurements are incorrect or no physical triangle can satisfy those constraints.