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Chapter 0: Problem 79

Give precise definitions of \(\csc ^{-1} x\) and \(\cot ^{-1} x\)

Short Answer

Expert verified
The \(\csc^{-1} x\) function is such that if \(y = \csc^{-1} x\), then \(x = \csc y\) and its domain and range are \(x\in\]-鈭�,-1]\cup\]1,鈭瀃\) and \(y\in\]0, 蟺/2]\cup\](蟺/2, 蟺]\) respectively. The \(\cot^{-1} x\) function is such that if \(y = \cot^{-1} x\), then \(x = \cot y\) and its domain is all real numbers and range is \(y\in\]0, 蟺[\).

Step by step solution

01

Define Inverse Cosecant

The inverse cosecant function, denoted \(\csc^{-1} x\), or arc-cosecant, is the function such that if \(y = \csc^{-1} x\), then \(x = \csc y\). The domain of \(\csc^{-1} x\) is \(x\in\]-鈭�,-1]\cup\]1,鈭瀃\) and the range is \(y\in\]0, 蟺/2]\cup\](蟺/2, 蟺]\). It's important to note that the cosecant function is not defined wherever sine function is 0.
02

Define Inverse Cotangent

The inverse tangent function, denoted \(\cot^{-1} x\) or arc-cotangent, is the function such that if \(y = \cot^{-1} x\), then \(x = \cot y\). The domain of \(\cot^{-1} x\) is all real numbers and the range is \(y\in\]0, 蟺[\). It's important to remember that the cotangent function is not defined wherever tangent function is 0.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inverse Cosecant Function
The inverse cosecant function, represented as \( \csc^{-1} x \), helps us find an angle when we know the cosecant of that angle. Understanding this function is key in trigonometry. If \( y = \csc^{-1} x \), then it implies \( x = \csc y \). This means that \( y \) is the angle whose cosecant is \( x \).

When using the inverse cosecant function, it's crucial to remember its domain and range:
  • Domain: \( x \in (-\infty, -1] \cup [1, \infty) \). This excludes values between -1 and 1 since cosecant can't take these values without involving imaginary numbers.
  • Range: \( y \in [-\frac{\pi}{2}, 0) \cup (0, \frac{\pi}{2}] \). This range includes angles excluding where sine is zero, because cosecant is undefined there.
These constraints ensure that \( \csc^{-1} x \) precisely indicates the correct angle, providing a unique solution for each input value.
Inverse Cotangent Function
The inverse cotangent function, denoted \( \cot^{-1} x \), is a useful tool for determining an angle when given its cotangent. The relationship is defined such that if \( y = \cot^{-1} x \), then \( x = \cot y \). Simply put, \( y \) is the angle whose cotangent is \( x \).

The properties of this function include:
  • Domain: All real numbers. This is broad, allowing any real number to be a valid input for inverse cotangent.
  • Range: \( y \in (0, \pi) \). This interval reflects the angles for which cotangent has defined values. It excludes integer multiples of \( \pi \), where the tangent function, and thus cotangent, is undefined.
Keeping these in mind while working with \( \cot^{-1} x \) ensures that the function can accurately and consistently calculate angles based on cotangent values.
Trigonometric Domains and Ranges
Understanding the domains and ranges of trigonometric functions is pivotal to their effective use. Trigonometric functions describe relationships between angles and ratios, but their inverse counterparts require specific domains and ranges to ensure they are well-defined and unambiguous.

  • Domains: In trigonometry, the domain refers to the set of input values (often angles) that can produce results. Each inverse trig function has a particular domain that ensures real, valid outputs. For example, \( \csc^{-1} x \) excludes the interval \((-1, 1)\) since cosecant can't reach these values without resulting in imaginary numbers.
  • Ranges: The range comprises possible outputs of a function. Each inverse function has a specified range fitting the classic unit circle's values. For instance, the range of \( \csc^{-1} x \) avoids spots where sine is zero to prevent invalid outcomes.
Grasping these concepts helps interpret and apply trigonometric functions properly, facilitating smoother problem-solving in mathematics and real-world contexts.

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