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The inverse cosecant function, denoted as \( \text{csc}^{-1}(x) \), is used to find the angle whose cosecant is a particular value \( x \). This function is crucial when dealing with trigonometric problems that require reversing the process of finding a cosecant from a given angle.
The inverse cosecant has a specific domain and range to ensure it operates correctly as a one-to-one function. The domain includes all real numbers \( x \) such that \(|x| \geq 1\).\( x \) cannot be any value between -1 and 1, because cosecant, being the reciprocal of sine, is undefined for such values.
The range of \( \text{csc}^{-1}(x) \) is the set of angles \([-\pi/2, 0) \cup (0, \pi/2]\). This range ensures that the output is always an angle whose cosecant exists (except integers for zero, where cosecant is undefined). This range strategically excludes \( 0 \) as \( \text{csc}(x) \) becomes undefined when it equals zero.

• The domain: \(|x| \geq 1\)
• The range: \([-\pi/2, 0) \cup (0, \pi/2]\)

The inverse secant function, denoted as \( \text{sec}^{-1}(x) \), reverses secant trigonometric values to find the corresponding angle. Like the inverse cosecant, the inverse secant has specified domains and ranges making it uniquely invertible for trigonometric calculations.
For \( \text{sec}^{-1}(x) \), the domain is defined as all real numbers \( x \) where \(|x| \geq 1\). This mirrors the limits of the inverse cosecant because secant and cosecant are both reciprocal functions of sine and cosine, respectively. Inside the interval, between -1 and 1, secant values do not exist because they are reciprocals of cosine which have values stretched beyond those limits.
The range of the inverse secant is the angle interval \([0, \pi/2) \cup (\pi/2, \pi]\). With this setting, the inverse secant function yields values that match the secant's reciprocals of cosine inputs, while still avoiding situations where secant would not be defined, such as \( \pi/2 \) where cosine is zero.

• The domain: \(|x| \geq 1\)
• The range: \([0, \pi/2) \cup (\pi/2, \pi]\)

The inverse cotangent function, represented as \( \text{cot}^{-1}(x) \), allows you to find the angle whose cotangent equals \( x \). Cotangent, the reciprocal of tangent, operates with its own distinct characteristics, which are reflected in its inverse.
Unlike inverse cosecant and secant, the inverse cotangent functions over all real numbers. Therefore, its domain is simply "all real numbers," which implies that for any real number \( x \), \( \text{cot}^{-1}(x) \) can find a corresponding angle.
The range of \( \text{cot}^{-1}(x) \) is from \( 0 \) to \( \pi \), represented as \((0, \pi)\). This range excludes the endpoints to avoid undefined behavior at \( 0\) and \(\pi\) where cotangent and tangent operations become undefined because the sine actually reaches zero.

• The domain: all real numbers
• The range: \((0, \pi)\)