91Ó°ÊÓ

The concept of absolute value is fundamental when solving trigonometric equations. In trigonometry, absolute value represents the distance of a number on the unit circle’s circumference from the origin, regardless of direction. For example, when we encounter an equation like \(|\cos x| = \frac{\sqrt{3}}{2}\), it implies that we are interested in all the angles where the cosine's value has a magnitude of \(\frac{\sqrt{3}}{2}\), regardless of whether it is positive or negative.

This understanding leads to two potential equations: one where the cosine of \(x\) is positive \(\left(\cos x = \frac{\sqrt{3}}{2}\right)\) and one where it is negative \(\left(\cos x = -\frac{\sqrt{3}}{2}\right)\). Since the cosine function can have both positive and negative values depending on the angle, we must consider both scenarios to find all possible solutions within the given interval.

Trigonometric functions are the relationships between the angles and sides of a triangle, particularly a right-angled triangle. They are also defined in terms of the unit circle, a circle with radius one centered at the origin of a coordinate system.

The main trigonometric functions are:

• Sine (sin)
• Cosine (cos)
• Tangent (tan)

In this exercise, we focus on the cosine function, which compares the distance along the adjacent side of the triangle to the hypotenuse in a right-angled triangle. However, when considering the unit circle, the cosine function gives the x-coordinate of a point on the circumference of the circle at a given angle from the positive x-axis. Understanding how these functions behave and their signs in different quadrants is essential to solving trigonometric equations effectively.

The principle of trigonometric cycles, also referred to as periodicity, refers to the repeating nature of trigonometric functions over specific intervals called periods. Each trigonometric function has a period over which the function values repeat. For the cosine function, its period is \(2\pi\), meaning that \(\cos(x)\) will have the same value every \(2\pi\) radians you move along the unit circle.

This property is crucial when solving equations like \(|\cos x| = \frac{\sqrt{3}}{2}\), as we must consider all angles that satisfy the equation within one full cycle of \(0\) to \(2\pi\). Using the periodic nature of the cosine function, we deduced that the negative solution gave us two additional angles, \(\frac{4\pi}{3}\) and \(\frac{5\pi}{3}\). These angles are found by recognizing that cosine is negative in the second and third quadrants of the unit circle, leading to the discovery of all solutions within the specified interval.