Problem 21 - 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. $$ a=42.1, c=37, A=112^{\circ} $$

Step-by-Step Solution

Answer

Upon solving, we find that with given measurements we have two possible triangles. For the first possible triangle the measurements are: \( B \approx 26^{\circ}\), \( C \approx 42^{\circ}\), and \( b \approx 23.8 \). For the second possible triangle, the measurements are: \( B \approx 26^{\circ}\), \( C \approx 138^{\circ}\), and \( b \approx 80.3 \).

Step 1: Examining the Given Angle

We first need to look at the given angle, which is Angle A. Is it acute or obtuse? In this case, it’s an obtuse angle because it measures \(112^{\circ}\). This means that this is the largest angle in the triangle, because no triangle can have more than one obtuse angle or angles totaling to more than \(180^{\circ}\).

Step 2: Apply the Sine Rule to Find Angle C

We can use the Law of Sines to find an angle of the triangle. Rearranged form of the Law of Sines is used: \( \frac{a}{\sin A} = \frac{c}{\sin C}\). Substituting the given values: \( \frac{42.1}{\sin 112^{\circ}} = \frac{37}{\sin C}\). Solve this equation to find \(\sin C\), then find \(C\) by taking inverse sine of \(\sin C\).

Step 3: Check the Possibilities of Triangle

If the sine of the calculated angle C using inverse sine lies between 1 and -1, then angle C exists and we have then one or two possible triangles. If it does not, then no triangle is possible. If angle C is acute, two triangles are possible: Triangle 1 with Angle C and Triangle 2 with angle \(180^{\circ} - C\). However, if angle C is obtuse, only one triangle is possible as a triangle cannot contain two obtuse angles. In this case, the sine value is within these bounds, and we do indeed have a valid angle, so a triangle is possible.

Step 4: Find the Remaining Angle and Side

To complete the triangle or triangles, we need to find the other angle by subtracting the known angles from \(180^{\circ}\). If two triangles were possible, we examine both possibilities with two different \(C\) angles. Once the third angle is known (B), we use the Law of Sines again to find the unknown side \( \frac{b}{\sin B} = \frac{a}{\sin A}\). Solve this equation to find the unknown side.

Key Concepts

SSA Condition

The SSA condition, also known as side-side-angle condition, refers to a scenario in which two sides and a non-included angle of a triangle are provided. This case can be intriguing as it does not always guarantee a unique solution for the triangle. It can result in:

• One triangle
• Two distinct triangles
• No triangle at all

When dealing with the SSA condition, it is essential to consider the measure of the given angle. If the angle is obtuse or right, then only one configuration is possible because the triangle cannot have another angle equal or greater. If the angle is acute, there is the potential for two valid triangles, referred to as the ambiguous case.
This is why SSA is sometimes humorously remembered by students as the "ambiguous case," as it can lead to these variable outcomes. Careful calculations need to be done to determine which, if any, triangles are valid under the given measurements.

Solving Triangles

Solving triangles involves determining all the unknown sides and angles of a triangle. In cases where two sides and a non-included angle are known, as in the SSA scenario, the process can be particularly challenging. The sequence typically includes:

• Identifying initial known elements such as sides and angles.
• Using trigonometric laws to determine unknown angles, such as applying the Law of Sines.
• Calculating the remaining sides using known angles and sides, again often with the Law of Sines.

Depending on the calculations, this procedure may indicate that there is no possible triangle, one unique triangle, or two possible triangles that satisfy the given conditions. Each calculation step is crucial to ensure proper understanding and identification of the solution's validity.

Triangle Geometry

Triangle geometry contains rules and properties specific to triangles. Each triangle must adhere to a few basic properties:

• The sum of all internal angles is always equal to 180 degrees.
• Only one angle can be obtuse.
• Each side must be less than the sum of the other two sides.

For the SSA scenario, these properties help in understanding potential outcomes. For example, knowing that a triangle cannot have more than one obtuse angle determines that if Angle A is obtuse, angles B and C must be acute. These geometric principles guide problem-solving approaches, ensuring logical deductions are made while solving the triangle.

Trigonometric Functions

Trigonometric functions play a vital role in solving triangles, especially under the SSA condition. The Law of Sines is a primary tool employed here: \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]This formula allows solving for unknown sides or angles when certain other sides or angles are known. The functions involve sine, cosine, and tangent, but in the context of the SSA case, the sine function is more frequently used. By rearranging this fundamental law, we can solve for unknowns effectively. To compute the angles, we use the inverse sine function, crucial when ensuring angle limits (like keeping values between -1 and 1). Handling these computations accurately enables precise solutions and dealing with the potential ambiguities presented by the SSA condition.