Problem 20 - 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. $$ a=57.5, c=49.8, A=136^{\circ} $$

Step-by-Step Solution

Answer

Given values can form one triangle. The values for angle B, C and side b must be calculated using the steps given.

Step 1: Determining the possibility of triangle formation

To determine the possibility of the triangle's existence, use the Sine Rule to find the ratio of the sides given to the sine of their opposite angles. Calculate \( \frac{a}{\sin A} \) and \( \frac{c}{\sin C} \). However, since \( A > 90^{\circ} \), there will only be one triangle.

Step 2: Finding Angle C

As the sum of angles in a triangle equals 180 degrees, calculate angle C using the formula \( C = 180^{\circ} - A - B \). However, we have yet to find Angle B. By the sine rule, we can deduce that sine B = \( \frac{b \sin A}{a} \). Once we find \( \sin B \), we can use it to find \( \sin C \) and from that value we can calculate angle C.

Step 3: Finding Side b

After acquiring the value for angle C, apply the law of sines again to find the value of side b. Using the formula \( b = \frac{a \sin B}{\sin A} \), the lengths of all three sides of the triangle, and the magnitude of all angles can be found.

Key Concepts

Sine Rule

The Sine Rule, also known as the law of sines, is a critical tool in solving triangles when two sides and a non-included angle are known, referred to as the SSA case (Side-Side-Angle). According to the Sine Rule, in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. Mathematically, this is expressed as:

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

This equation helps us find unknown sides or angles of a triangle when certain other pieces are already known. In the scenario of an SSA triangle, you can use the Sine Rule to solve for unknown angles or sides by employing the given measurements. This makes it a versatile and essential formula in trigonometry.

Angle Calculation

Calculating angles in a triangle is essential for triangle accuracy. Knowing that the total sum of angles in any triangle is 180 degrees provides a basis for finding unknown angles.

• Start by calculating one of the missing angles using the Sine Rule and given side lengths.
• Use the information that the sum of the interior angles of a triangle is \( 180^\circ \) to find the other unknown angle.

For instance, if angles A and B are known, angle C can easily be determined by:

• \( C = 180^{\circ} - A - B \)

This method ensures you can always find the missing angles once any two are known.

Law of Sines

The Law of Sines is a fundamental principle in trigonometry that helps in solving for unknowns in a triangle. It is particularly useful in situations where two sides and an opposite angle are given. This law not only aids in determining missing sides or angles, but also ensures the relationships between sides and angles are upheld.

• The Law states that the sides of a triangle are proportional to the sines of their opposite angles.

You apply this law to find missing values using known measurements. Once the missing elements are identified, confirming the correctness with the Law of Sines consolidates the solution. This method rectifies potential computational gaps, affirming the triangle's validity.

Triangle Existence Conditions

Understanding when a set of measurements can form a triangle is crucial. In the SSA (Side-Side-Angle) scenario, determining whether a triangle can exist hinges on specific criteria:

• The given angle should not exceed 180 degrees minus the sum of the other two sides' angles.
• Recognize that an angle greater than 90 degrees (an obtuse angle) usually restricts the formation to one possible triangle, especially when the side opposite it is the longest given side.

In the exercise provided, where angle \(A = 136^{\circ}\), there is a unique scenario - it ensures only one solution exists. This is because an angle above 90 degrees limits the triangle's geometry to right or acute angles for other vertices. Understanding triangle existence helps in avoiding unnecessary calculations or solving attempts when no triangle is feasible with given data.