Problem 25 - 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. $$ a=16, b=18, A=60^{\circ} $$

Step-by-Step Solution

Answer

The given measurements can form two different triangles. The first has angles A=60 degrees, B=70.6 degrees, C=49.4 degrees and sides a=16, b=18, c=13.0. The second has angles A=60 degrees, B=109.4 degrees, C=10.6 degrees and sides a=16, b=18, c=3.1.

Step 1: Find the Measure of Angle B

Using the Law of Sines, the measure of angle B can be calculated: \[ B = \sin^{-1}\left(\frac{b \cdot \sin(A)}{a}\right) \] \[ B = \sin^{-1}\left(\frac{18 \cdot \sin(60^\circ)}{16}\right) \]After doing the math, we find that \(B \approx 70.6^\circ\) or \(B = 180^\circ - 70.6^\circ = 109.4^\circ\]. Because the value of angle B is larger than 90 degrees but less than 180 degrees, there is another possible solution for angle B considering the properties of a triangle.

Step 2: Determine Number of Possible Triangles

If the sum of angles A, B and C (the third angle) is equal to 180 degrees, the measurements can form a triangle. Let's find angle C for both possible B values:- For \(B = 70.6^\circ\), \(C = 180^\circ - 60^\circ - 70.6^\circ = 49.4^\circ\)- For \(B = 109.4^\circ\), \(C = 180^\circ - 60^\circ - 109.4^\circ = 10.6^\circ\)In both cases, all angles are positive and less than 180 degrees, which indicates two possible triangles.

Step 3: Calculating the Third Side

The length of side c can be found using the Law of Sines on both possible triangles:- For \(B = 70.6^\circ\) and \(C = 49.4^\circ\), \(c = \frac{a \cdot \sin(C)}{\sin(A)} = \frac{16 \cdot \sin(49.4^\circ)}{\sin(60^\circ)} \approx 13.0\)- For \(B = 109.4^\circ\) and \(C = 10.6^\circ\), \(c = \frac{a \cdot \sin(C)}{\sin(A)} = \frac{16 \cdot \sin(10.6^\circ)}{\sin(60^\circ)} \approx 3.1\)

Key Concepts

Law of Sines

The Law of Sines is a powerful tool for solving triangles, particularly when we're given two sides and an angle (SSA). It relates the ratios of each side to the sine of its opposite angle. This law states that for any triangle, the following equation holds true:
\[\begin{equation}\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\end{equation}\]
This relationship allows us to find unknown angles and sides, given that we have enough information about the triangle. For example, if we know two sides and an angle that isn't between them (like in our exercise), we can use the Law of Sines to find the other angles, and eventually, the third side.

• Helpful tip: Always make sure the angle you're working with is opposite the side given to maintain the correct ratio.
• To solve for an angle, we rearrange the formula to isolate the angle and then use the inverse sine function.

Ambiguous Case of the Law of Sines

One of the most intriguing aspects of the Law of Sines is the ambiguous case. This occurs uniquely in SSA configurations, where two sides and a non-included angle are known. In this situation, different outcomes are possible: no triangle, one triangle, or two triangles. The ambiguity arises when the given angle is acute (less than 90 degrees), and the opposite side is shorter than the other given side but longer than the product of the other side and the sine of the given angle. That's why it's crucial to check for all possible scenarios when solving SSA triangles.
For our specific case, the exercise illustrates this ambiguity. The given angle A is 60 degrees, and since the side opposite it (side a) is shorter than side b, there are potentially two different values for angle B.

• An important calculation step is to determine whether the values of the sine function result in an angle that is logically possible within the triangle context.
• The measurement must also satisfy the condition that the sum of angles in a triangle is always 180 degrees.

SSA Triangle Solution

Solving SSA triangles can be straightforward once you've understood the Law of Sines and the ambiguous case it presents. After calculating one possible angle using the Law of Sines, you often have to consider the supplementary angle to account for the ambiguous case. In our exercise, after determining one value of angle B, we found its supplementary angle to see if another triangle is possible.
The next step involves checking that the angles do not exceed 180 degrees in total and that no angle is equal to or greater than 180 degrees. Once validated, the third side length can be calculated with the Law of Sines for each possible triangle. This comprehensive assessment ensures that we find all the possible solutions for an SSA triangle and accurately solve for their dimensions.

• It's essential to remember to round your final answers appropriately, as the exercise requires rounding sides to the nearest tenth and angles to the nearest degree.
• Visualization can significantly help to understand how the pieces fit together, so don't hesitate to sketch out the triangles as you work through them.