Problem 25 - 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. $$ A=33^{\circ}, a=2, b=3.5 $$

Step-by-Step Solution

Answer

There is no solution for this triangle.

Step 1: Identify the Type of Problem

The problem gives the measures of angle \( A = 33^\circ \), side \( a = 2 \), and side \( b = 3.5 \). This is an ambiguous case where we need to apply the Law of Sines to determine if there are one, two, or no possible triangles.

Step 2: Apply the Law of Sines

Use the Law of Sines to find \( \sin B \): \[ \frac{a}{\sin A} = \frac{b}{\sin B} \] Substitute the given values: \[ \frac{2}{\sin 33^\circ} = \frac{3.5}{\sin B} \] Calculate \( \sin B \): \[ \sin B = \frac{3.5 \cdot \sin 33^\circ}{2} \approx 0.9497 \] This implies \( \sin B > 1 \), which indicates there is no solution for this triangle, as sine values must be between -1 and 1.

Key Concepts

Law of Sines

The Law of Sines is a fundamental principle in trigonometry used to relate the angles and sides of a triangle. It is often particularly useful in solving triangles that are not right triangles. The law states:

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

This equation means the ratio of the length of a side of a triangle to the sine of the opposite angle is the same for all three sides and angles in the triangle.

In practical terms, if you know one angle and its opposite side, along with another side, you can use the Law of Sines to find the missing angle. In the provided exercise, we used the Law of Sines to determine \( \sin B \) by relating the known values of angle \( A \) and its opposite side \( a \) with the known side \( b \).

However, it's crucial to remember that the outputs for the sine of an angle should be between -1 and 1. If the calculation results in a value outside this range, it implies an invalid triangle. This concept leads us to the triangle's solution type as seen in the original exercise.

Angle-Side Relationships

Understanding the relationships between angles and sides in a triangle is crucial for determining whether a triangle is possible and, if so, its type. In any triangle, there are certain rules:

• The largest angle is always opposite the longest side.
• The smallest angle is opposite the shortest side.
• The sum of any two sides must be greater than the third side (Triangle Inequality Theorem).

In our example, the given angle and its opposite side along with another side were analyzed to determine if a triangle could exist. When you find, for instance, that \( \sin B \) is greater than 1, like in our exercise, it violates the sine value boundaries, indicating that such an angle is not feasible. This effectively tells us that the setup does not form a valid triangle.

Considering how side lengths align and how angles relate in a triangle helps in assessing the possibility of triangle formation, whether a triangle is acute, right, or obtuse, and even whether different types of triangle solutions exist.

Triangle Solution Types

In trigonometry, particularly when using the Law of Sines, you may encounter different possible solutions for a triangle. Here's what you should look for:

• No Solution: The side lengths and angles provided do not satisfy the conditions for forming a triangle. This occurs, as shown in the exercise, when a calculated sine exceeds the bounds of -1 to 1, suggesting the triangle cannot exist.
• One Solution: The angles and sides align such that exactly one distinct triangle is possible.
• Two Solutions: Known as the Ambiguous Case, this typically happens with given information like SSA (Side-Side-Angle) and can sometimes lead to two different but valid triangles.

In our example, the computed value \( \sin B \) was greater than 1 which meant there was no solution because such a sine value defies trigonometric principles, confirming there is no possible configuration of a triangle.

Being aware of these solutions helps in comprehending the ambiguity inherent in some triangles and ensures accurate resolution and analysis of triangle-related problems.