Understanding the Inverse Cosine Function: Exploring arccos(x), its Domain, and Range

arccos(x)

The symbol “arccos(x)” represents the inverse cosine function, also known as the arcsine function

The symbol “arccos(x)” represents the inverse cosine function, also known as the arcsine function. It is the inverse of the cosine function. This means that if you input a value “x” into the arccos function, it will give you the angle whose cosine is “x”.

The domain of arccos(x) is -1 ≤ x ≤ 1, since the cosine function only takes values within this range. The range of arccos(x) is 0 ≤ arccos(x) ≤ π (or 0 ≤ arccos(x) ≤ 180 degrees), representing the angle in radians (or degrees) between 0 and π (or 0 and 180 degrees).

To better understand the concept, let’s look at an example. Suppose we have arccos(0.5). To find the angle θ whose cosine is 0.5, we can use a calculator. Most calculators have a “Cos” button and an “Inverse” or “Second” button to find the inverse cosine.

Using the calculator, we can input arccos(0.5) or cos^(-1)(0.5) and obtain the result of approximately 60 degrees or π/3 radians. This means that the angle whose cosine is 0.5 is 60 degrees or π/3 radians.

It is important to note that the inverse cosine function has multiple solutions within its range. For example, if we input arccos(-0.5), we would get approximately 120 degrees or 2π/3 radians, since the cosine of this angle is also -0.5.

In summary, arccos(x) is the inverse function of the cosine function, used to find the angle whose cosine is equal to “x”. Its domain is -1 ≤ x ≤ 1, and its range is 0 ≤ arccos(x) ≤ π (or 0 ≤ arccos(x) ≤ 180 degrees).

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