Problem 28 - 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. $$ a=7, b=28, A=12^{\circ} $$

Step-by-Step Solution

Answer

Two triangles are possible. The measurements for the first triangle are \( B=77.5^{\circ}, C=90.5^{\circ}, c=29.2 \), and for the second triangle \( B'=102.5^{\circ}, C'=65.5^{\circ}, c'=17.1 \).

Step 1: Identify possible cases for SSA

In SSA case, we can have one of the three possibilities:\n1. If \( b \leq a \sin A \), no triangle is possible.\n2. If \( b > a \sin A \) and \( b < a \), one triangle is possible.\n3. If \( b \geq a \), two triangles are possible.\nNow, we have to check which case is applicable for the given values of \( a = 7 \), \( b = 28 \), and \( A = 12^{\circ} \).

Step 2: Check for existence of triangle

First calculate \( a \sin A \). In this case, we have \( 7 \sin 12^{\circ} = 1.46 \). Comparing with \( b = 28 \), we find \( b > a \sin A \). Now, compare \( b \) and \( a \). \( b > a \) is also true here. These comparisons show that we are dealing with the third case, i.e., two triangles are possible.

Step 3: Solve for remaining angles and side for the first triangle

Use the Law of Sines to find angle \( B \) as \( B = \sin^{-1} ((b \sin A) / a) = \sin^{-1} ((28 \sin 12^{\circ}) / 7) = 77.5^{\circ} \). Now find angle \( C \) using the fact that the sum of angles in a triangle is 180^{\circ}. So \( C = 180^{\circ} - A - B = 180^{\circ} - 12^{\circ} - 77.5^{\circ} = 90.5^{\circ} \). Lastly the side \( c \) can be found using the Law of Sines again: \( c = b \sin C / \sin B = 28 \sin 90.5^{\circ} / \sin 77.5^{\circ} = 29.2 \).

Step 4: Solve for remaining angles and side for the second triangle

The second possible triangle will have \( B' = 180^{\circ} - B = 180^{\circ} - 77.5^{\circ} = 102.5^{\circ} \). Similarly, we find \( C' = 180^{\circ} - A - B' = 180^{\circ} - 12^{\circ} - 102.5^{\circ} = 65.5^{\circ} \). Using the Law of Sines to find side \( c' = b \sin C' / \sin B' = 28 \sin 65.5^{\circ} / \sin 102.5^{\circ} = 17.1 \).

Key Concepts

Law of Sines

The Law of Sines is an important tool in trigonometry for solving triangles, especially those that cannot be addressed using basic geometry. When you are given two sides and an angle that is not between them (SSA condition), the Law of Sines can help determine unknown sides and angles in the triangle.
This law states that in any triangle:

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

Here, \(a\), \(b\), and \(c\) are the sides of the triangle, while \(A\), \(B\), and \(C\) are the angles opposite those sides, respectively.
To solve a triangle with SSA conditions, use the known values to find any unknowns:

• If side \( b \), side \( a \), and angle \( A \) are known, you can find angle \( B \) first.
• From there, use the fact that the angles in a triangle add up to \( 180^\circ \) to find angle \( C \).
• Lastly, you can use the Law of Sines again to find the third side, \( c \).

Exploring the Law of Sines lays the groundwork for understanding more complex cases, like the ambiguous case.

Ambiguous Case

The ambiguous case arises when solving triangles using the Law of Sines under SSA conditions. This situation can lead to one, two, or no possible triangles, making it a bit tricky.
There are three potential outcomes in the ambiguous case:

• No triangle if \( a \sin A \) is greater than or equal to \( b \).
• One triangle if \( b \) is greater than \( a \sin A \) and less than or equal to \( a \).
• Two triangles if \( b \) is greater than or equal to \( a \).

The key is comparing \( a \sin A \) with side \( b \).
In our specific problem, \( a = 7 \), \( b = 28 \), and \( A = 12^\circ \):

• First, calculate \( a \sin A = 7 \sin 12^\circ \approx 1.46 \).
• Notice that \( b = 28\), which is much larger than \( a \sin A \).

Since \( b > a \sin A \) and \( b \geq a \), there are two possible triangles in this scenario.

Triangle Solutions

After determining that two triangles are possible from the SSA condition, the next step is to solve both triangles fully.
For the first triangle:

• Calculate angle \( B \) using \( B = \sin^{-1} \left( \frac{b \sin A}{a} \right) = \sin^{-1} \left( \frac{28 \sin 12^\circ}{7} \right) \approx 77.5^\circ \).
• Then find angle \( C \) by understanding the sum of angles in a triangle: \( C = 180^\circ - A - B = 90.5^\circ \).
• Determine side \( c \) using the Law of Sines: \( c = \frac{b \sin C}{\sin B} \approx 29.2 \).

For the second triangle:

• Find angle \( B' \): \( B' = 180^\circ - B = 102.5^\circ \).
• Calculate \( C' \): \( C' = 180^\circ - A - B' = 65.5^\circ \).
• Solve for side \( c' \) with the Law of Sines: \( c' = \frac{b \sin C'}{\sin B'} \approx 17.1 \).

This thorough approach ensures you find all potential solutions, adhering to the principles of solving a triangle with an SSA set of information.