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Cosine is an even function

Prerequisites

Even functions | \(f(x) = f(-x)\)

Description

The identity \(\htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000024}{\theta}) = \htmlClass{sdt-0000000124}{\cos}(-\htmlClass{sdt-0000000024}{\theta})\) is a reflection of the fact that \( \htmlClass{sdt-0000000124}{\cos} \) is an even function. This identity highlights the symmetry of the cosine function about the vertical axis, meaning that the function's graph looks the same on both sides of the \( \htmlClass{sdt-0000000017}{y} \)-axis. It essentially indicates that the cosine function is independent of the direction of the angle on the unit circle, whether measured clockwise or counterclockwise from the positive \( \htmlClass{sdt-0000000003}{x} \)-axis. This property is crucial in trigonometry and is used in various mathematical and physical contexts to simplify the analysis of periodic and oscillatory phenomena. Understanding this identity helps in solving trigonometric equations and in the study of waves, vibrations, and alternating currents, where symmetrical properties around a central point are common.

\[\htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000024}{\theta}) = \htmlClass{sdt-0000000124}{\cos}(-\htmlClass{sdt-0000000024}{\theta})\]

Symbols Used:

This is a commonly used symbol to represent an angle in mathematics and physics.
\( \cos \)

This is the symbol for cosine, a trigonometric function that calculates the ratio of the adjacent side to the hypotenuse of a right-angled triangle.

Derivation

  1. On the unit circle, \( \htmlClass{sdt-0000000124}{\cos} \)(\( \htmlClass{sdt-0000000024}{\theta} \)) represents the \(x\)-coordinate of a point that is \( \htmlClass{sdt-0000000024}{\theta} \) radians around the circle from the positive \(x\)-axis. The angle \(-\htmlClass{sdt-0000000024}{\theta}\) represents rotating in the opposite direction by the same amount.
  2. Due to the symmetry of the circle, rotating \( \htmlClass{sdt-0000000024}{\theta} \) radians in the positive direction and \(-\htmlClass{sdt-0000000024}{\theta}\) radians in the negative direction lands on points that are reflections of each other across the \(x\)-axis. However, both points have the same \(x\)-coordinate because they are at the same horizontal level, just on opposite sides of the \(x\)-axis.
  3. Consider the definition of an even function:
    \[\htmlClass{sdt-0000000120}{f(x)} = \htmlClass{sdt-0000000096}{f}(-\htmlClass{sdt-0000000003}{x})\]
    The cosine function is an even function because the \(x\)-coordinate (or the value of \( \htmlClass{sdt-0000000124}{\cos} \)(\( \htmlClass{sdt-0000000024}{\theta} \)) is the same whether we move clockwise or counterclockwise from the positive \(\htmlClass{sdt-0000000003}{x}\)-axis by the same angle \( \htmlClass{sdt-0000000024}{\theta} \).
  4. Therefore, from the symmetry of the unit circle, the definition of even functions, and mathematical properties such as Euler's formula, we can derive and understand why \(\htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000024}{\theta}) = \htmlClass{sdt-0000000124}{\cos}(-\htmlClass{sdt-0000000024}{\theta})\), showcasing that the cosine function is even and does not change with the sign of its argument.

Example

Given \(\htmlClass{sdt-0000000024}{\theta} = \frac{\htmlClass{sdt-0000000126}{\pi}}{4}\) radians, we calculate:

\[\htmlClass{sdt-0000000124}{\cos}(\frac{\htmlClass{sdt-0000000126}{\pi}}{4}) = \frac{\sqrt2}{2}\] and

\[\htmlClass{sdt-0000000124}{\cos}(\frac{\htmlClass{sdt-0000000126}{\pi}}{4}) = \frac{\sqrt2}{2}\]

This demonstrates the fact that cosine is even