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The cosine function is a fundamental part of trigonometry and is widely used in mathematics to measure angles and lengths in right triangles and circles. It takes an angle as input and returns a value between -1 and 1. The cosine of an angle in a right triangle is the adjacent side divided by the hypotenuse. From a unit circle perspective, this translates to the horizontal coordinate of a point \((x, y)\) on the circle.

• Function characteristics: Cosine is an even function, meaning \(\cos(-x) = \cos(x)\).
• Range: It ranges from -1 to 1 for any angle \(x\), repeating its values every \(2\pi\) radians.
• Periodicity: The period of the cosine function is \(2\pi\), indicating its values repeat every \(2\pi\) radians.

The core property used in solving our exercise is that certain angles have specific cosine values, often derived from special triangles like the 45°-45°-90° triangle. This triangle has a cosine value of \(\frac{\sqrt{2}}{2}\), directly relevant for angles such as \(\frac{\pi}{4}\) and \(\frac{7\pi}{4}\) on the unit circle.

The unit circle is a helpful tool for understanding trigonometric functions, providing a bridge between geometry and algebra. It's a circle with a radius of 1, centered at the origin \((0, 0)\) on the Cartesian plane. Each point on the unit circle corresponds to an angle \(\theta\), measured from the positive x-axis.

• Angle and coordinates: An angle \(\theta\) intersects the unit circle at a point \((\cos \theta, \sin \theta)\).
• Cosine on the circle: The x-coordinate of these points gives the cosine value of the angle.
• Quadrants: Angles \(\theta\) can be measured as positive (counterclockwise) or negative (clockwise) from the x-axis, impacting the sign of the cosine value depending on the quadrant.

Using the unit circle helps determine exact values for trigonometric functions, such as identifying angles where \(\cos(\theta) = \frac{\sqrt{2}}{2}\) at \(\theta = \frac{\pi}{4}\) and \(\theta = \frac{7\pi}{4}\). This visualization aids in solving equations by directly connecting angles with their cosine values through symmetry and periodicity.

The general solution of a trigonometric equation addresses all possible values of the variable that satisfy the equation. Trigonometric functions, such as cosine, are periodic, repeating their values in a regular cycle. The general solution encompasses this periodic nature.Consider our core problem: solving \(\cos x = \frac{\sqrt{2}}{2}\). We find specific angles, \(x = \frac{\pi}{4}\) and \(x = \frac{7\pi}{4}\), where the cosine yields this value. Due to the periodicity of the cosine function, these angles repeat every \(2\pi\) radians.

• Periodic addition: To account for all such occurrences, add multiples of the period: \(x = \frac{\pi}{4} + 2k\pi\) and \(x = \frac{7\pi}{4} + 2k\pi\), where \(k\) is an integer.
• Infinite solutions: The general solution captures an infinite set of angles, achieved by varying the integer \(k\).

The concept of a general solution is crucial in trigonometry and beyond, allowing us to account for every instance of a repeating pattern within a function. This framework applies broadly across mathematical problems where periodic behavior is evident.