Problem 24 - Algebra 2 and Trigonometry

Question

Use the formula cos \(C=\frac{a^{2}+b^{2}-c^{2}}{2 a b}\) to show that the measure of each angle of an equilateral triangle is \(60^{\circ} .\)

Step-by-Step Solution

Answer

Each angle of an equilateral triangle is indeed \(60^\circ\).

Step 1: Understanding the Problem

In an equilateral triangle, all sides are equal. Let's denote the sides of the triangle as \(a\). Therefore, the formula given becomes \(\cos C = \frac{a^2 + a^2 - a^2}{2a \cdot a} = \frac{a^2}{2a^2}\). We need to show that this equals \(\cos(60^\circ)\).

Step 2: Expressing the Formula

Since all sides of an equilateral triangle are equal, substitute \(a = b = c\) in the formula: \\cos C = \frac{a^2 + a^2 - a^2}{2a \, a}\. Simplifying this, \\cos C = \frac{a^2}{2a^2}\.

Step 3: Simplifying the Expression

Further simplify the expression. Since the numerator and denominator both have \(a^2\), they cancel out, leaving: \\cos C = \frac{1}{2}\.

Step 4: Relating to Known Values

From trigonometry, we know that \\cos(60^\circ) = \frac{1}{2}\. Therefore, \\cos C = \frac{1}{2}\ implies angle \(C\) is \(60^\circ\).

Step 5: Concluding the Solution

Since angle \(C\) was arbitrary and any angle in the equilateral triangle could serve the same role due to the triangle's symmetry, each angle in the triangle measures \60^\circ\.

Key Concepts

Equilateral Triangle

An equilateral triangle is a special type of triangle where all three sides are of equal length. This property leads to some interesting features:

• Each angle in an equilateral triangle measures 60 degrees.
• The triangle is also equiangular, which means all three internal angles are equal.
• An equilateral triangle can also be considered as a regular polygon with three sides.

The symmetry of the equilateral triangle makes calculations simpler, especially when dealing with trigonometric functions and identities.
For instance, if you use the formula for cosine in any of its angles, you will always find that the cosine of the angle is equivalent to cos(60°), which is 1/2.

Trigonometric Identity

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables. In the case of the equilateral triangle,
we used a specific identity for the cosine of an angle. The key formula we analyzed is:\[ \cos C = \frac{a^2 + b^2 - c^2}{2ab} \]When this formula is applied to an equilateral triangle, since all the sides are equal (i.e., \(a = b = c\)), it simplifies to:\[ \cos C = \frac{a^2}{2a^2} = \frac{1}{2} \]This reveals that the angle in question is 60 degrees, due to the fact that \(\cos 60^\circ = \frac{1}{2}\). This identity not only serves as a proof but also reinforces a fundamental property of the equilateral triangle.

Angle Measurement

Understanding angle measurement is crucial when solving geometric problems or using trigonometric identities. An angle is measured in degrees or radians, with 360 degrees making a full circle.
In context of an equilateral triangle, we are dealing with degrees. Each angle in an equilateral triangle measures precisely 60 degrees:

• The property of equal sides in an equilateral triangle dictates equal angles.
• Using trigonometric identities, we derive this measurement, confirming consistencies with known trigonometric values.
• Knowing one angle helps in determining the interior structure of the triangle without additional calculations.

These measurements are not only essential in geometry, but also form the basis for more complex mathematical concepts.