Problem 7 - College Trigonometry

Question

Solve for the remaining side(s) and angle(s) if possible. As in the text, \((\alpha, a)\), \((\beta, b)\) and \((\gamma, c)\) are angle-side opposite pairs. $$ \alpha=68.7^{\circ}, a=88, b=92 $$

Step-by-Step Solution

Answer

\(\beta \approx 75.9^{\circ}, \gamma \approx 35.4^{\circ}, c \approx 55.0\)

Step 1: Determine the Triangle Type

Given two sides and an angle not included between them (SSA configuration), start by identifying that this might be the ambiguous case of the law of sines (SSA). This can lead to zero, one, or two solutions for the triangle.

Step 2: Apply the Law of Sines

Use the Law of Sines to find angle \(\beta\):\[ \frac{\sin(\alpha)}{a} = \frac{\sin(\beta)}{b} \]Substitute the given values:\[ \frac{\sin(68.7^{\circ})}{88} = \frac{\sin(\beta)}{92} \]Solve for \(\sin(\beta)\):\[ \sin(\beta) = \frac{92 \cdot \sin(68.7^{\circ})}{88} \]

Step 3: Calculate \(\sin(\beta)\) and Check Feasibility

Calculate \(\sin(\beta)\):\[ \sin(\beta) \approx \frac{92 \cdot 0.9272}{88} = 0.9695 \]Since \(\sin(\beta)\) is less than 1, it is feasible to find \(\beta\).

Step 4: Solve for Angle \(\beta\)

Use the inverse sine function to find \(\beta\):\[ \beta = \sin^{-1}(0.9695) \approx 75.9^{\circ} \]

Step 5: Calculate Remaining Angle \(\gamma\)

Use the angle sum of a triangle to find \(\gamma\):\[ \gamma = 180^{\circ} - \alpha - \beta \]\[ \gamma = 180^{\circ} - 68.7^{\circ} - 75.9^{\circ} = 35.4^{\circ} \]

Step 6: Find the Remaining Side \(c\)

Apply the Law of Sines again to solve for side \(c\):\[ \frac{\sin(\gamma)}{c} = \frac{\sin(\alpha)}{a} \]Substitute \(\gamma = 35.4^{\circ}\) and solve for \(c\):\[ c = \frac{a \cdot \sin(\gamma)}{\sin(\alpha)} = \frac{88 \cdot \sin(35.4^{\circ})}{\sin(68.7^{\circ})} \]Calculate \(c\):\[ c \approx \frac{88 \cdot 0.5796}{0.9272} = 55.0 \]

Key Concepts

Triangle Classification

When working with triangles, it is fundamental to classify them based on their sides and angles. Triangles have a variety of classifications:

• Equilateral: All sides and angles are equal.
• Isosceles: Two sides and two angles are equal.
• Scalene: All sides and angles are different.
• Acute: All angles are less than 90°.
• Right: One angle is exactly 90°.
• Obtuse: One angle is more than 90°.

In this specific exercise, we are dealing with a Scalene and Acute triangle, given different side lengths and only acute angles (less than 90°). This classification helps us decide which problem-solving strategies and trigonometric rules might apply.

Ambiguous Case SSA

The ambiguous case arises in a triangle when two sides and a non-included angle are provided. This configuration is often abbreviated as SSA (side-side-angle). It's called 'ambiguous' because such information can result in:

• Zero triangles, if the given side is shorter than the height of the triangle.
• One triangle, if the side is the shortest and only one possible height exists.
• Two triangles, if both shorter and longer side solutions are possible.

In the scenario of this exercise, the given values (\(\alpha=68.7^{\circ}, a=88, b=92\) ) fit the SSA ambiguous case. After applying the Law of Sines, we verified that one feasible triangle can be formed. Understanding the cases ensures we approach solving with the correct expectation.

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to find angles when the sine, cosine, or tangent values are known. In the case of this exercise, the inverse sine function (\(\sin^{-1}\)) was used. It transformed the trigonometric value back into an angle. This step is crucial when working with the Law of Sines, as the result can determine if a single solution exists or if the information possibly suggests multiple scenarios.For instance, finding \(\beta\) required:

• Calculating \(\sin(\beta)\)
• Using \(\sin^{-1}(0.9695)\) to find the actual angle, approximately 75.9°.

This function offers a bridge from numerical results back to angular measurements, essential in solving trigonometric problems.

Angle Sum Property of Triangles

The angle sum property is a foundational concept in triangle solutions. It states that the sum of the interior angles in any triangle is always 180 degrees. Knowing two angles allows you to easily find the third, ensuring that all angles add up correctly.In our exercise:

• Given \(\alpha = 68.7^{\circ}\) and \(\beta = 75.9^{\circ}\), the angle \(\gamma\) was calculated as \(180^{\circ} - 68.7^{\circ} - 75.9^{\circ} = 35.4^{\circ}\).

This principle not only helps verify calculations but also ensures the feasibility of the solution when dealing with ambiguous scenarios. Proper application confirms all angles are valid for triangle solutions, offering one more check in problem-solving.