Problem 26 - Algebra and Trigonometry

Question

\(19-28\) . Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. $$ a=100, \quad b=80, \quad \angle A=135^{\circ} $$

Step-by-Step Solution

Answer

One triangle: \(\angle B \approx 31.72^{\circ}, \angle C \approx 13.28^{\circ}, c \approx 32.44\).

Step 1: Identify Known Values

We are given the side lengths \(a = 100\), \(b = 80\), and angle \(\angle A = 135^{\circ}\). We need to find \(\angle B\), \(\angle C\), and side \(c\).

Step 2: Apply the Law of Sines

The Law of Sines states: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). Using \( \frac{a}{\sin A} = \frac{b}{\sin B} \), plug in known values: \( \frac{100}{\sin 135^{\circ}} = \frac{80}{\sin B} \).

Step 3: Solve for \(\sin B\)

Calculate \(\sin 135^{\circ}\).\[ \sin 135^{\circ} = \sin (180^{\circ} - 45^{\circ}) = \sin 45^{\circ} = \frac{\sqrt{2}}{2} \]Substitute to find \(\sin B\):\[ \frac{100}{\frac{\sqrt{2}}{2}} = \frac{80}{\sin B} \]\[ \sin B = \frac{80 \times \frac{\sqrt{2}}{2}}{100} = \frac{40\sqrt{2}}{100} = \frac{2\sqrt{2}}{5} \]

Step 4: Determine Possible \(\angle B\) Values

Calculate \(\angle B\) using \(\sin^{-1}\).\[ \angle B = \sin^{-1} \left( \frac{2\sqrt{2}}{5} \right) \approx 31.72^{\circ} \]Due to the ambiguous case (SSA condition), consider a second possible angle:\[ \angle B = 180^{\circ} - 31.72^{\circ} = 148.28^{\circ} \]

Step 5: Find Corresponding \(\angle C\)

For \(\angle B = 31.72^{\circ}\), calculate \(\angle C\):\[ \angle C = 180^{\circ} - 135^{\circ} - 31.72^{\circ} = 13.28^{\circ} \]For \(\angle B = 148.28^{\circ}\), calculate \(\angle C\):\[ \angle C = 180^{\circ} - 135^{\circ} - 148.28^{\circ} = -103.28^{\circ} \] (invalid, since angles must be positive).

Step 6: Calculate Length of \(c\)

Use the Law of Sines \(\frac{a}{\sin A} = \frac{c}{\sin C}\) for \(\angle B = 31.72^{\circ}\):\[ \frac{100}{\frac{\sqrt{2}}{2}} = \frac{c}{\sin 13.28^{\circ}} \]\[ c = 100 \times \frac{\sin 13.28^{\circ}}{\frac{\sqrt{2}}{2}} \approx 32.44 \]

Step 7: Conclusion

Only one triangle is valid with \(\angle B = 31.72^{\circ}\) and \(\angle C = 13.28^{\circ}\). The side \(c \approx 32.44\).

Key Concepts

Solving Triangles

Solving triangles involves finding unknown sides and angles of triangles when at least some values are known. This can be accomplished using various mathematical rules and properties, such as the Law of Sines and the Law of Cosines. Triangles can have different configurations such as scalene, isosceles, and equilateral, which influence how we approach solving them.

When using the Law of Sines to solve a triangle, the equation \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \) is a great starting point. This law applies to all types of triangles and can be particularly useful when certain combinations of sides and angles are known, like in the case of angle-side-side (ASS) or side-side-angle (SSA).

Ultimately, solving triangles is about piecing together the information using mathematical principles and careful calculation. Knowing how to apply the correct rule and accurately handle the ambiguity can lead to finding all possible triangle configurations that meet the given conditions.

Ambiguous Case

The ambiguous case occurs in the context of the Law of Sines when using the SSA condition. This scenario arises because given two sides and a non-included angle, there might be two different triangles that satisfy these conditions. This occurrence is what makes solving triangles sometimes tricky.

To identify an ambiguous case:

• Check if you are given two sides and a non-included angle (SSA).
• Use the Law of Sines to solve for a possible angle.

After calculating the first possible angle, the supplementary angle (i.e., \(180^\circ\) minus the found angle) should also be considered. This is because the sine of an angle and its supplement are equal. However, you should remember that the sum of angles in any triangle is always \(180^\circ\), and any angle exceeding this will invalidate the triangle.

In our exercise, the original angle B resulted in 31.72°, leading to a valid triangle. The supplementary angle resulted in 148.28°, which, when summed with the given angle \(135^\circ\), exceeded \(180^\circ\), making it an invalid scenario.

Triangle Angles

Triangle angles are crucial in understanding how the parts of a triangle relate to each other. Every triangle has three angles, and the sum of these angles is always \(180^\circ\). This property is used extensively in solving triangle problems as it allows mathematicians to solve for unknown angles when the others are known.

When using the Law of Sines and once an angle is found, you can easily find the remaining angles of the triangle using the above property. However, it is critical to ensure angles stay within the permissible limits, i.e., angles must be positive and less than \(180^\circ\).

• Always verify if angles result in a valid triangle by adding them.
• Account for the possibility of error in the ambiguous case by calculating all viable angle options.

Diamond shapes or squares cannot form in normal triangle situations. Precision in angle assessments guarantees the correct triangle configuration is found.

Mathematics Problem Solving

Mathematics problem solving is the procedure of using logical reasoning and mathematical concepts to solve problems. Solving a triangle is a clear illustration of this process. This involves setting up equations, strategically using known values, and systematically arriving at unknowns.

Here are basic steps in mathematics problem solving:

• Understand the problem by identifying known and unknown values.
• Select the appropriate mathematical formulas needed for the problem.
• Use logical reasoning to connect the dots and solve for unknowns.
• Verify solutions by plugging them back into the original equations.

Problem solving isn't solely about finding correct regions; it’s about understanding why those solutions make sense. Applying theory such as the Law of Sines to find angles and sides is foundational in ensuring accurate outcomes. Exercising these principles with patience and diligence allows for refining one's skills and understanding in mathematics.