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Inequalities are a crucial concept in mathematics that help in comparing the size or order of two quantities. They often involve the symbols '<', '>', '≤', or '≥'. In the context of trigonometric functions like \(\sin x\) and \(\cos x\), inequalities express the constraint that their values must stay within a certain range due to the properties of these functions.
For \(\sin x\) and \(\cos x\), an important inequality is \(-1 \leq \sin x \leq 1\) and \(-1 \leq \cos x \leq 1\).
This means that no matter what value of \(x\) you choose, the output of the sine and cosine functions will always lie between \(-1\) and \(1\). These inequalities derive from the unit circle concept, where any point has a distance of 1 from the center, hence limiting the value to a maximum of 1 either positively or negatively.

The absolute value of a number is essentially its distance from zero on the number line, without considering direction. It is a way to measure size without regard to whether the number is positive or negative.
For example, the absolute value of -3 is 3, written as \(|-3| = 3\).
When we consider the absolute value in the context of trigonometric functions, it simplifies how we express inequalities.
Given the inequalities \(-1 \leq \sin x \leq 1\) and \(-1 \leq \cos x \leq 1\), applying the absolute value results in \(|\sin x| \leq 1\) and \(|\cos x| \leq 1\).

• This tells us that the maximum amplitude—without considering negative values—is capped at 1.
• It removes the need to separately consider negative and positive parts of the inequality, as the absolute value already accounts for both.

The unit circle is a fundamental tool in trigonometry, offering a simple and effective way to understand sine and cosine functions. A unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane.
This circle helps visualize the values of trigonometric functions at various angles. Here’s how it works:

• The horizontal coordinate of a point on the circle corresponds to \(\cos x\).
• The vertical coordinate corresponds to \(\sin x\).

Given that every point on the unit circle is at a distance of 1 from the origin, the maximum and minimum values for \(\sin x\) and \(\cos x\) are 1 and -1, respectively.
The unit circle not only simplifies calculation but also enables intuitive understanding of how these functions behave as you move around the circle. As you travel around it, the sine and cosine values oscillate smoothly between -1 and 1. These visualizations provide clear proof of why \(|\sin x| \leq 1\) and \(|\cos x| \leq 1\) for all real numbers \(x\).