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The cosine function is a fundamental trigonometric function that relates an angle in a right triangle to the ratio of the adjacent side to the hypotenuse.
It's also a critical function in understanding wave patterns, rotations, and oscillations.
The general form of the cosine function is \(\cos(x)\).
Key points to remember about the cosine function include:

• It oscillates between -1 and 1.
• The function is periodic with a period of \(2\pi\).
• Important values include \(\cos(0) = 1\), \(\cos(\pi/2) = 0\), and \(\cos(\pi) = -1\).

When solving an equation like \(\cos(x) = \frac{\sqrt{2}}{2}\), it's helpful to memorize the special angles and their cosine values from the unit circle.
For the given problem, recall that \(\cos(\pi/4) = \frac{\sqrt{2}}{2}\) and this same cosine value occurs at \(7\pi/4\) (or equivalently \(-\pi/4\)).

The unit circle is a circle of radius 1 centered at the origin in a coordinate plane.
It is a powerful tool for understanding trigonometric functions and their values at various angles.
The relationship between the angle and the coordinates on the unit circle can be represented as \(\cos(\theta)\) for the x-coordinate and \(\sin(\theta)\) for the y-coordinate.
Key elements of the unit circle include:

• It has a radius of 1 unit.
• The angle \(\theta\) is measured from the positive x-axis.
• Each point on the circle is of the form \( (\cos(\theta), \sin(\theta)) \).

For solving trigonometric equations, angles that result in specific cosine values are found by tracing the x-coordinates on the unit circle.
For the problem \(\cos(x) = \frac{\sqrt{2}}{2}\), the angles that correspond to this cosine value on the unit circle are at \(\pi/4\) and \(7\pi/4\).

Periodic functions are functions that repeat their values in regular intervals or periods.
In trigonometry, both sine and cosine functions are periodic with a period of \(2\pi\).
This means that for any angle \(\theta\), \(\cos(\theta) = \cos(\theta + 2k\pi)\) where \(k\) is any integer. Important characteristics of periodic functions include:

• The repeating pattern, known as the period.
• The amplitude, which is the maximum value the function attains.
• The phase shift, which can move the function left or right along the x-axis.

When solving equations involving periodic functions like \(\cos(x) = \frac{\sqrt{2}}{2}\), the periodic nature means there are multiple solutions.
For our problem, the general solutions are \(x = \frac{\pi}{4} + 2k\pi\) and \(x = \frac{7\pi}{4} + 2k\pi\).
This accounts for all possible angles that satisfy the original equation within a full cycle of the cosine function.