Problem 76 - A Graphical Approach to Precalculus with Limits

Question

Give the exact real number value of each expression. Do not use a calculator. $$\csc \left(\csc ^{-1} 3\right)$$

Step-by-Step Solution

Answer

The value is 3.

Step 1: Understand the Function

The function given is \(\csc(\csc^{-1}(x))\). The function \(\csc^{-1}(x)\) is the inverse of the cosecant function, which means \(\csc^{-1}(x)\) gives the angle whose cosecant is \(x\). Similarly, \(\csc(\theta) = x\).

Step 2: Apply Inverse Properties

For the function \(y = \csc^{-1}(x)\), it implies that \(\csc(y) = x\). Therefore, \(\csc(\csc^{-1}(x)) = x\). This property holds where \(x\) is in the domain of the inverse cosecant function, which is \(|x| \geq 1\).

Step 3: Substitute the Expression

For \(\csc(\csc^{-1}(3))\), by the property of the inverse function, it simplifies directly to \(3\) without further calculation, as long as the value \(3\) is within the permissible domain \(|x| \geq 1\).

Key Concepts

Cosecant

The cosecant function, often denoted as \( \csc \theta \), is one of the six fundamental trigonometric functions. It is related to other trigonometric functions like sine, cosine, and tangent. The cosecant of an angle \( \theta \) is defined as the reciprocal of the sine function: \( \csc \theta = \frac{1}{\sin \theta} \).
Understanding the cosecant function is essential in trigonometry, as it allows us to solve various geometric problems by providing relationships between angles and side lengths. Unlike sine, which has a range of \([-1, 1]\), the cosecant function has no minimum or maximum sheltering. This is because as a reciprocal function, it will approach infinity or negative infinity when the sine approaches zero (which is why \( \csc \theta \) is undefined when \( \sin \theta = 0 \)).
The cosecant function is periodic with a period of \(2\pi\), just like sine. However, unlike sine, it has vertical asymptotes every time the sine function is zero, aligning with its undefined nature at those points. Understanding these properties is crucial when assessing the behavior and application of the cosecant in different trigonometric contexts.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the involved variables. They are incredibly useful in simplifying expressions and solving trigonometric equations. The concept of inverse functions plays a pivotal role here. When we say \( \csc^{-1}(x) \), we mean the angle whose cosecant is \(x\). The identities that employ inverse functions can often significantly simplify complex expressions.
A fundamental identity involving the cosecant and its inverse is \( \csc(\csc^{-1}(x)) = x \). This reflects how inverse trigonometric functions allow us to unravel initial trigonometric expressions back to their raw numerical values, provided that \(x\) falls within the domain constraints of the function.
Understanding these identities allows students to not only solve problems like \( \csc(\csc^{-1}(3)) \) effortlessly but also to evaluate, analyze, and infer the properties of trigonometric functions in broader mathematical and real-world applications.

Domain and Range

The domain and range of a trigonometric function painted the limits within which the function operates and produces output. For cosecant, the function is defined as the reciprocal of sine, which introduces specific guidelines on its domain. The sine function yields values between \(-1\) and \(1\), so its reciprocal, cosecant, is not defined within \((-1, 1)\) — meaning for \( \csc \theta \) to be valid, \( \sin \theta \) must be either \(1\) or \(-1\), or any value lower or higher than this range can encompass.
Cosecant's domain is all real numbers except where sine is zero, that is, integer multiples of \( \pi \), resulting in exclusions at points like \(0, \pi, 2\pi, \) etc. For its inverse, \( \csc^{-1}(x) \), the domain is \(|x| \geq 1\).
By confirming that the expression \(3\) fits within this domain, we can safely apply the inverse property, knowing that calculating \( \csc(\csc^{-1}(3)) \) will lead us directly back to \(3\). This reassures users that they are working within the correct functionality scope, providing confidence in resolving similar problems.