Problem 24 - A Graphical Approach to Precalculus with Limits

Question

Solve triangle. \(B=52^{\circ}, C=29^{\circ}, a=43\) centimeters

Step-by-Step Solution

Answer

A = 99°, b ≈ 34.03 cm, c ≈ 21.08 cm.

Step 1: Find Angle A Using Sum of Angles in a Triangle

The sum of the angles in any triangle is always \(180^{\circ}\). To find angle \(A\), use the formula: \(A = 180^{\circ} - B - C\). Substitute the given values: \(A = 180^{\circ} - 52^{\circ} - 29^{\circ}\). Calculate \(A = 99^{\circ}\).

Step 2: Calculate Side b Using the Law of Sines

Use the law of sines to find side \(b\): \(\frac{a}{\sin A} = \frac{b}{\sin B}\). Substitute the known values into the equation: \(\frac{43}{\sin 99^{\circ}} = \frac{b}{\sin 52^{\circ}}\). Solve for \(b\) by multiplying both sides by \(\sin 52^{\circ}\): \(b = 43 \cdot \frac{\sin 52^{\circ}}{\sin 99^{\circ}}\). Calculate \(b\approx 34.03\) centimeters.

Step 3: Calculate Side c Using the Law of Sines

Use the law of sines again to find side \(c\): \(\frac{a}{\sin A} = \frac{c}{\sin C}\). Substitute the known values into the equation: \(\frac{43}{\sin 99^{\circ}} = \frac{c}{\sin 29^{\circ}}\). Solve for \(c\) by multiplying both sides by \(\sin 29^{\circ}\): \(c = 43 \cdot \frac{\sin 29^{\circ}}{\sin 99^{\circ}}\). Calculate \(c \approx 21.08\) centimeters.

Key Concepts

Sum of Angles in a Triangle

Triangular geometry is based on a simple and crucial rule: the sum of all internal angles in a triangle is always equal to \(180^{\circ}\). This is a fundamental principle that helps us solve for unknowns in triangles.
In the given exercise: you are tasked with finding angle \(A\). This becomes easy by subtracting the known angles \(B\) and \(C\) from \(180^{\circ}\):

• The formula used is \(A = 180^{\circ} - B - C\).
• Substituting the given values \(B = 52^{\circ}\) and \(C = 29^{\circ}\): \(A = 180^{\circ} - 52^{\circ} - 29^{\circ}\).
• Which results in \(A = 99^{\circ}\).

This rule is an essential part of triangle problem-solving because it ensures we can find that missing angle whenever two angles are known.

Law of Sines

The Law of Sines is a powerful tool in trigonometry for solving triangles. It finds unknown sides or angles by relating the ratios of each angle to its opposite side. This law states: \[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\] In our problem, this law lets us find unknown sides \(b\) and \(c\) from the known side \(a = 43\) and angles.

• To find side \(b\), rearrange the equation: \(b = 43 \cdot \frac{\sin 52^{\circ}}{\sin 99^{\circ}}\)
• Similarly, solve for \(c\): \(c = 43 \cdot \frac{\sin 29^{\circ}}{\sin 99^{\circ}}\)

By using the Law of Sines, these calculations become straightforward, and we achieve accurate results for both sides.

Angle Calculation

When solving triangles, the calculation of unknown angles applies key geometric principles.
In the task, we already knew angles \(B = 52^{\circ}\) and \(C = 29^{\circ}\), and we needed the third angle \(A\).

• The principle used was the sum of angles in triangles: \(A + B + C = 180^{\circ}\)
• By rearranging this, \(A = 180^{\circ} - B - C\), simplifying direct calculation.

Calculating angles precisely using these known mathematical rules simplifies understanding and solving more complex triangle scenarios down the line.

Side Calculation

Finding side lengths in triangles is often achieved using the Law of Sines.
In this exercise, after determining angle \(A = 99^{\circ}\), and knowing side \(a = 43\), we found sides \(b\) and \(c\):

• To determine \(b\), we plugged values into the rearranged Law of Sines equation: \(b = 43 \cdot \frac{\sin 52^{\circ}}{\sin 99^{\circ}}\), resulting in \(b \approx 34.03\) cm
• For \(c\), we used: \(c = 43 \cdot \frac{\sin 29^{\circ}}{\sin 99^{\circ}}\), resulting in \(c \approx 21.08\) cm

Using the Law of Sines facilitates finding unknown side lengths when angles and at least one side are known. This is crucial for accurate triangle solutions.