Problem 32 - A Graphical Approach to Precalculus with Limits

Question

Solve triangle. \(C=71.83^{\circ}, B=42.57^{\circ}, a=2.614\) centimeters

Step-by-Step Solution

Answer

The triangle's angles are \(A=65.6^{\circ}\), \(B=42.57^{\circ}\), and \(C=71.83^{\circ}\), and sides are \(a=2.614\) cm, \(b \approx 1.855\) cm, \(c \approx 2.723\) cm.

Step 1: Calculate Angle A

In a triangle, the sum of the interior angles is always \(180^{\circ}\). Given \(C = 71.83^{\circ}\) and \(B = 42.57^{\circ}\), we can find \(A\) by subtracting the sum of \(B\) and \(C\) from \(180^{\circ}\):\[A = 180^{\circ} - C - B = 180^{\circ} - 71.83^{\circ} - 42.57^{\circ} = 65.6^{\circ}\]

Step 2: Use the Law of Sines to Find Side b

The Law of Sines states that for any triangle, \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\). We can use this to solve for side \(b\):\[\frac{2.614}{\sin 65.6^{\circ}} = \frac{b}{\sin 42.57^{\circ}}\]Solving for \(b\), we compute:\[b = \frac{2.614 \times \sin 42.57^{\circ}}{\sin 65.6^{\circ}} \approx 1.855 \text{ cm}\]

Step 3: Use the Law of Sines to Find Side c

Again, using the Law of Sines, we solve for side \(c\):\[\frac{2.614}{\sin 65.6^{\circ}} = \frac{c}{\sin 71.83^{\circ}}\]Solving for \(c\), we compute:\[c = \frac{2.614 \times \sin 71.83^{\circ}}{\sin 65.6^{\circ}} \approx 2.723 \text{ cm}\]

Key Concepts

Angle Calculation

Calculating angles in a triangle is a fundamental step towards solving and understanding its properties. In any triangle, the sum of the interior angles is always exactly 180 degrees. This property is a basic rule you can rely on to find an unknown angle when the other two are known.

For instance, in our exercise, we have been given two angles, \(C = 71.83^{\circ}\) and \(B = 42.57^{\circ}\). To find the third angle, \(A\), we simply subtract the sum of the known angles from 180 degrees: \(A = 180^{\circ} - 71.83^{\circ} - 42.57^{\circ} = 65.6^{\circ}\).

This method of angle calculation allows you to solve for missing angles easily and is a key tool in the study of triangle properties.

Law of Sines

The Law of Sines is a powerful tool used in solving triangles, particularly when you are dealing with non-right triangles. It provides a relationship between the sides of a triangle and its angles through the equation: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\).

This law is particularly useful when you know some sides and angles and need to find others, like in our exercise.

• First compute side \(b\) using \(b = \frac{2.614 \times \sin 42.57^{\circ}}{\sin 65.6^{\circ}} \approx 1.855\) cm.
• Next, calculate side \(c\) via \(c = \frac{2.614 \times \sin 71.83^{\circ}}{\sin 65.6^{\circ}} \approx 2.723\) cm.

Each segment of the calculation utilizes the given side and angles, demonstrating the flexibility of the Law of Sines in finding unknown triangle parts.

Triangle Properties

Triangles are fascinating figures with unique properties that make them versatile in geometry and real-world applications. One key property is that their internal angles always sum to 180 degrees. This is foundational to understanding angle calculations.

Additionally, the ability to use the Law of Sines across various triangle forms shows their adaptability and the intricate relation between their angles and sides. Knowing a few angles and sides allows you to find missing measurements while maintaining these essential relationships.

• An isosceles triangle has at least two equal sides and angles, helping determine unknown sides.
• A scalene triangle, where all sides have different lengths, requires the application of such laws as the Law of Sines for solving.

The maintenance of such properties in various types of triangles underscores how deeply geometry is ingrained in our problem-solving processes.