Cosine Periodicity

The trigonometric identity that describes the periodicity of the cosine function is \(\htmlId{tooltip-cosine}{\cos}(\htmlId{tooltip-angle}{\theta}) = \htmlId{tooltip-cosine}{\cos}(\htmlId{tooltip-angle}{\theta} + 2\htmlId{tooltip-pi}{\pi} \htmlId{tooltip-integer}{k}), \htmlId{tooltip-integer}{k} \in \mathbb{Z}\) , where \( \htmlId{tooltip-angle}{\theta} \) is the angle in radians and \( \htmlId{tooltip-integer}{k} \) is any integer. This identity signifies that the cosine function is periodic with a period of \(2\htmlId{tooltip-pi}{\pi}\), meaning that the function repeats its values every \(2\htmlId{tooltip-pi}{\pi}\) radians. It reflects the fundamental characteristic of the cosine function, which is its cyclical nature as it oscillates between \(−1\) and \(1\). The identity is crucial for solving trigonometric equations and understanding the behavior of waves and oscillations in various fields such as physics, engineering, and mathematics. It also underpins the analysis of harmonic motion and the study of wave functions in quantum mechanics, showcasing the widespread applicability of trigonometric periodicity.

1. On the unit circle(see figure below), the cosine of an angle \( \htmlId{tooltip-angle}{\theta} \) is defined as the \( \htmlId{tooltip-var}{x} \)-coordinate of the point where the line forming the angle with the positive \( \htmlId{tooltip-var}{x} \)-axis intersects the circle. Since the unit circle has a radius of 1, this definition is consistent for all angles.
2. When you rotate a point on the unit circle by \(2\htmlId{tooltip-pi}{\pi}\) radians (a full circle), the point returns to its original position. This means that adding \(\htmlId{tooltip-pi}{\pi}\) radians to an angle \( \htmlId{tooltip-angle}{\theta} \) results in a point with the same x-coordinate as the starting point, due to the circular nature of the unit circle.
3. Mathematically, this concept can be expressed as \(\htmlId{tooltip-cosine}{\cos}(\htmlId{tooltip-angle}{\theta}) = \htmlId{tooltip-cosine}{\cos}(\htmlId{tooltip-angle}{\theta} + 2\htmlId{tooltip-pi}{\pi} \htmlId{tooltip-integer}{k})\), where \( \htmlId{tooltip-integer}{k} \) is any integer. Adding \(2\htmlId{tooltip-pi}{\pi} \htmlId{tooltip-integer}{k}\) to \( \htmlId{tooltip-angle}{\theta} \) effectively rotates the point \( \htmlId{tooltip-integer}{k} \) full circles around the unit circle, which does not change its \( \htmlId{tooltip-var}{x} \)-coordinate.

Suppose we choose \(\htmlId{tooltip-angle}{\theta} = \frac{\htmlId{tooltip-pi}{\pi}}{4}\) radians and \(\htmlId{tooltip-integer}{k}=1\). This choice of \( \htmlId{tooltip-integer}{k} \) indicates that we are looking one full cycle ahead from our initial angle. According to the identity, the cosine of this angle plus \(2\htmlId{tooltip-pi}{\pi}\) times \( \htmlId{tooltip-integer}{k} \) should be equal to the cosine of the initial angle.

Therefore, we have:

\[\htmlId{tooltip-cosine}{\cos}(\frac{\htmlId{tooltip-pi}{\pi}}{4}) = 2\htmlId{tooltip-pi}{\pi} \cdot 1 = \htmlId{tooltip-cosine}{\cos}(\frac{\htmlId{tooltip-pi}{\pi}}{4})\]

The cosine of \(\frac{\htmlId{tooltip-pi}{\pi}}{4}\) is \(\frac{\sqrt(2)}{2}\)​. Let's verify that \(\htmlId{tooltip-cosine}{\cos}(\frac{\htmlId{tooltip-pi}{\pi}}{4} = 2\htmlId{tooltip-pi}{\pi}\) yields the same result. Since adding \(2\htmlId{tooltip-pi}{\pi}\) to an angle results in a full circle rotation, the angle effectively remains the same in terms of its position on the unit circle, thus preserving the cosine value. Let's calculate it to confirm.

The calculation confirms that \(\htmlId{tooltip-cosine}{\cos}(\frac{\htmlId{tooltip-pi}{\pi}}{4}) = 0.7071067811865476\) and \(\htmlId{tooltip-cosine}{\cos}(\frac{\htmlId{tooltip-pi}{\pi}}{4} + 2\htmlId{tooltip-pi}{\pi}) = 0.7071067811865477)\). The slight difference in the last decimal place is due to rounding errors in the computation, but essentially, both values are equal to \(\frac{\sqrt(2)}{2}\)​​, as expected. This numerical example illustrates the periodicity of the cosine function, demonstrating that adding \(2\htmlId{tooltip-pi}{\pi}\) (a full circle rotation) to the angle does not change the value of the cosine function, consistent with the identity \( \htmlId{tooltip-cosine}{\cos} \)(\( \htmlId{tooltip-angle}{\theta} \)) = \( \htmlId{tooltip-cosine}{\cos} \)(\( \htmlId{tooltip-angle}{\theta} \) + 2\( \htmlId{tooltip-pi}{\pi} \) \( \htmlId{tooltip-integer}{k} \)).