Problem 6 - 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. $$ A=123^{\circ}, a=12, b=23 $$

Step-by-Step Solution

Answer

No solution because \( \sin B > 1 \) is not possible.

Step 1: Analyze Given Information

We are given a triangle with angle \( A = 123^{\circ} \), side \( a = 12 \), and side \( b = 23 \). We need to determine if the triangle can have no solution, one solution, or two solutions, and then solve it.

Step 2: Use Law of Sines

We use the law of sines: \( \frac{a}{\sin A} = \frac{b}{\sin B} \). Substituting the given values, we have \( \frac{12}{\sin 123^{\circ}} = \frac{23}{\sin B} \).

Step 3: Solve for \( \sin B \)

Rearrange to get \( \sin B = \frac{23 \times \sin 123^{\circ}}{12} \). Calculate \( \sin 123^{\circ} \approx 0.8387 \), thus \( \sin B \approx \frac{23 \times 0.8387}{12} \approx 1.6074 \).

Step 4: Determine Number of Solutions

Since \( \sin B > 1 \), this is impossible in the context of real triangles (\( \sin \theta \) must be between -1 and 1). This means there is no solution as \( B \) can't be an angle in a triangle.

Key Concepts

Triangle Solutions

When solving triangles, one of the primary tasks is to determine if a particular triangle configuration can represent a real, physical triangle. Triangles follow certain geometric rules. Sometimes, based on the given angles and sides, a triangle may have:

• No solution: the given measurements do not form a real triangle.
• One solution: the measurements fit the criteria exactly, giving one specific triangle.
• Two solutions: there are two distinct triangles possible with the same set of measurements.

In this case, we used the Law of Sines to attempt to solve the triangle with a given angle measure of 123°, and side lengths of 12 and 23. By checking the value of \( \sin B \), calculated from the law, and finding that it's greater than 1, we determined that a triangle cannot exist. This happens because the sine function only takes values between -1 and 1 for real angles, indicating no possible triangle can have an angle with such a sine value.

Sine Function

The sine function is a fundamental concept in trigonometry and it describes the relationship between an angle in a right triangle and the ratio of the length of the side opposite the angle to the length of the hypotenuse. It is represented mathematically as:\[ \\sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}} \\]The function is periodic, with values ranging between -1 and 1. When determining \( \sin B \) from the Law of Sines for the triangle, we used the formula\[ \\frac{a}{\sin A} = \frac{b}{\sin B} \\]and solved for \( \sin B \). In an unexpected case where \( \sin B > 1 \), it indicates an error in triangle formation, showing that the angles and sides do not match up to form a valid triangle. This highlights the importance of understanding the sine function's range and its implications for solving triangles.

Trigonometry

Trigonometry is the branch of mathematics dealing with the relationships between the angles and sides of triangles. It's essential in solving triangles whether they are right triangles or oblique triangles. Key trigonometric functions include sine, cosine, and tangent. For solving triangles, the Law of Sines is a critical tool. It applies to any triangle:

• The formula \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \) relates the sides and angles.
• This law helps determine unknown angles and sides given adequate initial information.
• If the result gives an impossible sine value (outside of -1 to 1), it indicates that the triangle cannot exist.

A deep understanding of trigonometry not only assists in solving geometry problems but also plays a vital role in various real-world applications, such as physics and engineering projects.