Problem 18 - 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. $$ a=30, b=20, A=50^{\circ} $$

Step-by-Step Solution

Answer

The given measurements can form one triangle. The solution for the triangle is approximately: \( A = 50^{\circ}, B = 30^{\circ}, C = 100^{\circ}, a = 30, b = 20, c =49.38 \).

Step 1: Analyzing the given

The given parameters are: side a = 30, side b = 20, and angle A = 50 degrees. It is important to note that the given angle A is opposite side a, and the other given side b is not adjacent to angle A.

Step 2: Check for the possibility of a triangle

Use the Sine rule to check whether the measurements can form a triangle. The Sine rule states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides and angles of the triangle. Apply this rule by comparing the known ratios: \( \frac{a}{\sin(A)} \) and \( b \). If \( b \) is less than \( a * \sin(A) \), two triangles could be possible. If \( b \) is equal to \( a * \sin(A) \), only one triangle can be formed. If \( b \) is greater than \( a * \sin(A) \), no triangle will be formed.

Step 3: Apply the Sine Rule

Calculate \( b = a * \sin(A) \) = 30 * sin(50) = 22.94 approximately. As \( b \) (which is 20) is less than 22.94, the measurements can form one or two triangles.

Step 4: Compute other angles

Calculate the angle opposite to side b using the Sine Rule. It can be acute (\( \arcsin(\frac{b}{a} * \sin(A)) \)) or obtuse (180 - \( \arcsin(\frac{b}{a} * \sin(A)) \)). The acute angle is \( B = \arcsin(\frac{20}{30} * \sin(50)) \) = 30 degrees (rounded). The obtuse angle is 150 degrees.

Step 5: Compute for one or two triangles

First, form one triangle using the acute angle for B (30 degrees). Calculate angle C using \( C = 180 - A - B \), which gives 100 degrees. Now, calculate side c using the Sine Rule (\( c = a * \frac{\sin(C)}{\sin(A)} \)). This gives one possible triangle: \( A = 50^{\circ}, B = 30^{\circ}, C = 100^{\circ}, a = 30, b = 20, c =49.38 \) (values rounded). Next, try to form a second triangle using the obtuse angle for B (150 degrees). But this is not possible as the sum of angles in a triangle should not exceed 180 degrees.

Key Concepts

Sine Rule

The Sine Rule, also known as the Law of Sines, is a principle in trigonometry that provides a relationship between the sides and angles of any triangle. Specifically, it states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant for all three sides and angles in the triangle. This rule is mathematically expressed as:
\[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \]
Where \( a, b, \) and \( c \) are the lengths of the sides of the triangle, and \( A, B, \) and \( C \) are the opposite angles, respectively. When given two sides and a non-included angle (SSA), the Sine Rule can help in determining the possibility of a triangle's existence. It also assists in calculating unknown angles or sides of the triangle when needed.

Using the Sine Rule in SSA Configuration

In an SSA scenario, the Sine Rule can help us assess whether a triangle is possible and, if so, whether there is one unique solution or two possible solutions. Once we've determined the feasibility of the triangle, we can then use the Sine Rule to solve for the unknowns. It becomes particularly useful in an exercise where we have two sides and an angle opposite to one of them, as is common in many geometry problems.

Law of Sines

The Law of Sines establishes a fundamental relationship between the sides and angles of a non-right triangle. This law is crucial in solving for unknown sides and angles, especially in cases of SSA (Side-Side-Angle) triangles. As a strategy for solving SSA triangle problems, one would first check if a triangle is possible by comparing the given side with the product of the other side and the sine of the given angle.
When dealing with the SSA configuration, it's essential to recognize the ambiguous case, which refers to the situation where two different triangles could be formed with the given measurements. The Law of Sines provides a method to resolve this ambiguity.
To illustrate the use of the Law of Sines, consider the ratio notation: \[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} \]
You can rearrange the terms to solve for the unknown side or angle. For example, to find angle \( B \) when sides \( a \) and \( b \), and angle \( A \) are known, you could use the formula:\[ B = \arcsin\left(\frac{b}{a} \cdot \sin(A)\right) \]
Given this information, you can determine whether one or two triangles are possible, or if no triangle can exist. This guides us in understanding the triangle's properties and solving for its parameters.

Determining Possible Triangles

When given two sides and an angle of a triangle (SSA), there are three possible outcomes. We can either form a single unique triangle, two different triangles, or no triangle at all. This determination is based on the given lengths of the sides and the specific angle, often resulting in the need for a careful analysis known as the 'ambiguous case' in the study of triangles.

Single Triangle Condition

If the length of given side \( b \) is precisely equal to \( a \sin(A) \), there's only one triangle that can be formed because side \( b \) forms a right angle with side \( a \). Similarly, if side \( b \) is less than \( a \sin(A) \) but greater than the height when side \( a \) acts as the base, one triangle can also be formed.

Two Triangles Condition

When side \( b \) is less than \( a \sin(A) \) and also greater than the opposite side \( a \), there are two possible triangles. This happens because there are two different angles that could correspond to the given side \( b \): one acute and one obtuse, leading to two potential configurations.

No Triangle Condition

If side \( b \) is greater than \( a \sin(A) \) and is therefore the longest side but opposite a non-right acute angle, no triangle can exist because the sides do not meet to form a complete figure.
Determining the possible number of triangles in an SSA scenario is crucial before proceeding to solve the triangle since the approach to finding the missing angles and side lengths will depend on the specific case presented. Careful analysis using the Sine Rule or Law of Sines makes this possible and ensures the correct interpretation and solutions of triangle problems.