Problem 26 - Algebra 2

Question

Determine whether each triangle has no solution, one solution, or two solutions. Then solve each triangle. Round measures of sides to the nearest tenth and measures of angles to the nearest degree. $$ A=68^{\circ}, a=3, b=5 $$

Step-by-Step Solution

Answer

There is no solution because \( \sin B \approx 1.547 \) is not possible.

Step 1: Identify the Given Components

The triangle has the following given components: Angle \( A = 68^\circ \), side \( a = 3 \), and side \( b = 5 \). We need to determine if there are no solutions, one solution, or two solutions by using the given angle-side-side (ASS) configuration.

Step 2: Applying the Law of Sines

Using the Law of Sines, we have \( \frac{a}{\sin A} = \frac{b}{\sin B} \). Substitute the known values: \( \frac{3}{\sin 68^\circ} = \frac{5}{\sin B} \). Calculate \( \sin B \) by rearranging: \( \sin B = \frac{5 \cdot \sin 68^\circ}{3} \approx 1.547 \).

Step 3: Analyzing \( \sin B \)'s Value

The value \( \sin B = 1.547 \) is not possible because the sine of an angle cannot exceed 1. This indicates there is no solution for this triangle with the given measures.

Key Concepts

Law of Sines

When exploring triangles in trigonometry, the Law of Sines is a powerful tool that helps us find unknown sides or angles. It relates the ratios of the lengths of sides of a triangle to the sines of its angles. The formula is expressed as:\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]where \(a\), \(b\), and \(c\) are the lengths of the sides opposite to angles \(A\), \(B\), and \(C\) respectively. The Law of Sines is particularly useful in situations where you know:

• Two angles and one side (AAS or ASA scenario)
• Two sides and a non-included angle (ASS scenario)

This last situation, known as the "ambiguous case", may lead to no solution, one solution, or two solutions. This ambiguity arises because the sine function can yield the same value for two different angles.

Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles, particularly right triangles. It's foundational for understanding how angle measures and side lengths interact, using functions like sine, cosine, and tangent.
These functions help translate angle measurements into ratios of side lengths, and vice versa. In the context of solving triangles, trigonometric rules and functions allow us to calculate unknown angles and sides when certain elements of a triangle are known.
Understanding these principles is vital for solving real-world problems, such as calculating heights or distances that are not directly measurable. The Law of Sines, which belongs to this mathematical discipline, is specifically tailored for non-right triangles, making trigonometry applicable beyond geometrical shapes.

Triangle Solutions

In solving triangles, especially when given certain constraints, it's essential to determine whether the configuration allows for no solution, one solution, or two solutions. This determination depends largely on the conditions of the given problem, particularly when using trigonometric laws like the Law of Sines.
In the original exercise, for example, with \(A = 68^{\circ}\), \(a = 3\), and \(b = 5\), the aim was to find \(B\) using the Law of Sines. Despite setting up the equation properly, \(\sin B\) was calculated as 1.547, which is impossible because sine values can only range from -1 to 1. This clearly indicated that no triangles exist with the given values, leading to a conclusion of no solutions.

• If \(\sin B\) were a valid value (between 0 and 1), a triangle might exist.
• A value close to 1 could lead to two potential solutions due to the ambiguous case.

Thus, solving triangles requires not only calculation but also analysis to interpret these results correctly.