Problem 13 - Calculus Single Variable

Question

Calculate the value of the given inverse trigonometric function at the given point. $$ \operatorname{arccsc}(-\sqrt{2}) $$

Step-by-Step Solution

Answer

The value of \( \operatorname{arccsc}(-\sqrt{2}) \) is \( -\frac{\pi}{4} \).

Step 1: Understand the Function

The function \( \operatorname{arccsc}(x) \) represents the inverse of the cosecant function. This means \( y = \operatorname{arccsc}(x) \) if and only if \( \csc(y) = x \). We are asked to find \( y \) such that \( \csc(y) = -\sqrt{2} \).

Step 2: Recall the cosecant and sine relationship

Recall that \( \csc(y) = \frac{1}{\sin(y)} \), thus \( \csc(y) = -\sqrt{2} \) implies \( \sin(y) = -\frac{1}{\sqrt{2}} \).

Step 3: Simplify the sine function

We rewrite \( \sin(y) = -\frac{1}{\sqrt{2}} \) as \( \sin(y) = -\frac{\sqrt{2}}{2} \). This is a standard angle for sine.

Step 4: Identify the angle

The angles where \( \sin(y) = -\frac{\sqrt{2}}{2} \) are \( y = -\frac{\pi}{4} \) and \( y = \frac{5\pi}{4} \). However, \( \operatorname{arccsc}(x) \) is typically limited to the range \( [-\frac{\pi}{2}, \frac{\pi}{2}] \), excluding zero.

Step 5: Choose the correct angle

The only value in the given range for the inverse cosecant function that satisfies \( \sin(y) = -\frac{\sqrt{2}}{2} \) is \( y = -\frac{\pi}{4} \).

Key Concepts

Cosecant Function

The cosecant function, abbreviated as \( \csc \), is the reciprocal of the sine function. Simply put, it is defined as \( \csc(y) = \frac{1}{\sin(y)} \). If you have the sine of an angle, you can easily calculate its cosecant by taking the reciprocal. The cosecant function is not as commonly used as the sine function, but it plays an important role in trigonometry, especially in the context of inverse functions.
The range of the cosecant function is all real numbers except for the interval \([-1, 1]\), since it encompasses the reciprocals of sine values that lie outside of this range. When dealing with inverse trigonometric functions such as \( \operatorname{arccsc}(x) \), we need to be mindful of the restricted range of the cosecant function.

Sine Function

The sine function, denoted by \( \sin \), is one of the fundamental trigonometric functions. It gives the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. The sine function is defined for all real numbers and has a range of \([-1, 1]\).
In the context of this exercise, the sine function is related to the cosecant function through the identity \( \csc(y) = \frac{1}{\sin(y)} \). When we know that \( \csc(y) = -\sqrt{2} \), we can deduce that \( \sin(y) = -\frac{1}{\sqrt{2}} \), simplifying to \( \sin(y) = -\frac{\sqrt{2}}{2} \). This is a typical value that corresponds to standard angles in the unit circle. When solving inverse trigonometric problems, it's crucial to remember these standard values.

Angle Identification

Identifying the angle that matches a given sine value involves understanding the unit circle and the range of various trigonometric functions. Specifically, for this exercise, we need to find an angle \( y \) where \( \sin(y) = -\frac{\sqrt{2}}{2} \).
This sine value is familiar and suggests two possible standard angles: \( y = -\frac{\pi}{4} \) and \( y = \frac{5\pi}{4} \), based on the unit circle. However, since \( \operatorname{arccsc}(x) \) has a principal range from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), excluding zero, we choose \( y = -\frac{\pi}{4} \) as it lies within the valid range.
Using angle identification correctly is key in solving trigonometric equations, especially when dealing with inverse functions that have specific domain restrictions.