Problem 29 - Algebra and Trigonometry

Question

In Exercises 17–32, two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results. Round to the nearest tenth and the nearest degree for sides and angles, respectively. $$ a=22, c=24.1, A=58^{\circ} $$

Step-by-Step Solution

Answer

Angles for triangle 1 are approximately: A = 58 degrees, B = 44 degrees, C = 78 degrees. Sides are: a = 22, b = 13.4, c= 24.1. Angles for triangle 2 are approximately: A = 58 degrees, B = 20 degrees, C = 102 degrees. Sides are: a = 22, b = 8.4, c= 24.1. Hence, two triangles can be formed.

Step 1: Solving Triangle for Possibilities

Firstly, calculate \( \sin(C) \) using the formula \( \frac{a}{\sin(A)} = \frac{c}{\sin(C)} \), which simplifies to \( \sin(C) = \frac{c \cdot \sin(A)}{a} \). Plug the given values in to get \( \sin(C) \approx 0.97 \)

Step 2: Check Triangle Possibility

The sine of an angle is between -1 and 1. As the calculated \( \sin(C) \) is within this range, there is a possibility for a triangle. If \( \sin(C) \) is greater than 1, there is no triangle, if it equals to 1, it's a right triangle.

Step 3: Check for Another Triangle Possibility

There can be two triangles. The reference angle for the sine of 0.97 is approximately 78 degrees. The other angle with the same sine is \( 180^{\circ} - 78^{\circ} \approx 102^{\circ} \). So, there can be another triangle when the angle C is approximately 102 degrees

Step 4: Solving Triangle 1

The third angle B for the triangle 1 can be found using the rule that the sum of the angles in a triangle equals to \( 180^{\circ} \). So, \( B = 180^{\circ} - A - C = 180^{\circ} - 58^{\circ} - 78^{\circ} \approx 44^{\circ} \). Then calculate the side b using the law of sines formula \( b = \frac{c \cdot \sin(B)}{\sin(C)} \approx 13.4 \)

Step 5: Solving Triangle 2

The third angle B for the triangle 2 can be found similarly to Triangle 1. This time \( B = 180^{\circ} - A - C = 180^{\circ} - 58^{\circ} - 102^{\circ} \approx 20^{\circ} \). Then calculate the side b using the law of sines formula \( b = \frac{c \cdot \sin(B)}{\sin(C)} \approx 8.4 \)

Key Concepts

Law of Sines

The law of sines is a fundamental concept used to solve triangles, particularly when you know two sides and an angle opposite one of those sides. This principle is presented as \( \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \). It connects the side lengths of a triangle with their opposite angles.

Using this law is crucial when handling the SSA condition (Side-Side-Angle), allowing us to find unknown sides or angles. To apply it, you find \( \sin(C) \) by rearranging the equation to \( \sin(C) = \frac{c \cdot \sin(A)}{a} \). Plug in the values to solve for the unknown angle or side.

• Use the known side (e.g., \(a\) or \(c\)) and its opposite angle (\(A\) or \(C\)), along with the law of sines, to find an unknown opposite side or angle.
• Focus on how each side-angled pair relates across a triangle, which is essential for solving or confirming possible multiple triangle scenarios in SSA.

Understanding the law of sines makes solving complex problems more approachable and helps visualize triangle geometry effectively.

Angle-Side-Side Condition

The angle-side-side (SSA) condition can be a tricky situation when solving triangles. Unlike other conditions, SSA does not always yield a unique solution, and it can sometimes lead to more than one triangle being possible.

When you solve a triangle with SSA, it means you know two sides and an angle that is not included between the known sides. Given this, the law of sines becomes crucial for finding possible solutions.

• This scenario, known as the "ambiguous case," might lead to one, two, or no triangles as possible solutions, because the sine function has two angles between 0° and 180° that have the same sine value.
• The calculated \(\sin(C)\), if within [-1, 1], indicates a potential for at least one triangle formation. Further investigation might reveal a second angle C, found as \(180^\circ - C\), which could lead to a second triangle if other conditions are met.

Being aware of this condition's intricacies helps in addressing ambiguities logically with specific calculations.

Triangle Possibilities

When working with triangles, especially under the SSA condition, determining the number of possible triangles is crucial. Depending on the measurements you have, it can result in one triangle, two triangles, or no triangle at all.

Here is how you evaluate:

• Calculate \( \sin(C) \) and check if it lies between -1 and 1. This tells you if a triangle solution is geometrically possible.
• If \( \sin(C) \) equals 1 or is less than 1 but more than 0, then a triangle is possible. If \( \sin(C) \) equals 1, you have a right triangle.
• If \( \sin(C) \) is less than 0 or more than 1, then constructing a triangle with the given sides and angles is impossible.
• With \( \sin(C)\) valid in range, explore if two solutions exist: one angle \(C\) and its supplementary angle \(180^\circ - C\). This can yield two potential triangles if both result in positive internal angles and consistent solutions within the triangle angle sum rule.

Understanding these triangle possibilities helps you logically determine the likelihood of multiple solutions and how to validate them.