Problem 61 - Precalculus

Question

Use the Law of sines to prove that all angles in an equilateral triangle must have the same measure.

Step-by-Step Solution

Answer

All angles in an equilateral triangle are \(60^\circ\).

Step 1: Understanding Equilateral Triangle

An equilateral triangle is a triangle in which all three sides are equal in length. We denote the sides as \(a\), \(b\), and \(c\); therefore, \(a = b = c\). Our goal is to show that all the angles of the triangle are equal.

Step 2: Applying the Law of Sines

The Law of Sines states that for a triangle with sides \(a\), \(b\), and \(c\), and angles \(A\), \(B\), and \(C\) opposite those sides, \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\). For an equilateral triangle, we have \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\). Since \(a = b = c\), we conclude \(\sin A = \sin B = \sin C\).

Step 3: Concluding the Equivalence of Angles

In a triangle, if \(\sin A = \sin B = \sin C\), then \(A = B = C\) because for a specific angle in \(0, \pi)\), the sine function is injective. Thus, in our equilateral triangle, all angles must be equal. Since a triangle's angles sum to \(180^\circ\), each angle must be \(60^\circ\).

Key Concepts

Equilateral Triangle

An equilateral triangle is a unique type of triangle where all three sides have equal length. If you picture any triangle, imagine that each side is exactly the same.
This is the hallmark of an equilateral triangle. Because of this equal side length, it has some interesting properties. One of these is that all the angles are also equal.

• Each angle in an equilateral triangle measures 60 degrees.
• The sides are usually denoted as \( a = b = c \).
• This equality of angles and sides makes constructions and calculations involving equilateral triangles both simple and elegant.

Knowing that these characteristics repeat for any equilateral triangle is crucial for understanding geometry and trigonometry principles.

Triangle Angles

Angles are a fundamental concept when it comes to understanding triangles. Each triangle, regardless of type, always adheres to the principle that the sum of its angles equals 180 degrees.
This is an essential geometric truth.

• In an equilateral triangle, because the sides are equal, the angles must also be equal.
• If all angles are equal, then each angle must be \( \frac{180^\circ}{3} = 60^\circ \).

Understanding this helps visualize how every triangle's angles relate to each other. For equilateral triangles, this means each corner measures exactly the same. This uniformity in angles is due to the triangle's equal sides, reinforcing each angle's equality.
Recognizing this concept helps keep angle calculations consistent and accurate.

Sine Function

The sine function is an essential tool in trigonometry that relates angles in triangles to side lengths. In any triangle with angles \( A \), \( B \), and \( C \), opposite sides are traditionally denoted as \( a \), \( b \), and \( c \) respectively.
The Law of Sines uses the sine function to establish a relationship between these angles and their opposite sides.

• The Law of Sines states: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). It's a profound connection that exists because of how the sine of an angle correlates with the lengths across from it.
• In an equilateral triangle, this relationship is straightforward since the sides are all equal, meaning \( a = b = c \).
• Hence, \( \sin A = \sin B = \sin C \), leading us to conclude that \( A = B = C \).

Understanding the sine function and its role in the Law of Sines helps reveal why all angles in an equilateral triangle are equal. It simplifies proving angle equality by showing that equal side lengths naturally enforce equal angles.