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Magazine Chapter 10: Problem 118
Assume that the range of arcsecant is \(\left[0, \frac{\pi}{2}\right) \cup\left[\pi, \frac{3 \pi}{2}\right)\) and that the range of arccosecant is \(\left(0, \frac{\pi}{2}\right] \cup\left(\pi, \frac{3 \pi}{2}\right]\) when finding the exact value. \(\operatorname{arccsc}\left(\csc \left(\frac{9 \pi}{8}\right)\right)\)
Short Answer
Expert verified
\( \operatorname{arccsc} \left( \csc \left( \frac{9 \pi}{8} \right) \right) = \frac{7 \pi}{8} \).
Step by step solution
01
Understand the Problem
The task is to find the value of \( \operatorname{arccsc} \left( \csc \left( \frac{9 \pi}{8} \right) \right) \). The function \( \operatorname{arccsc} \) returns the angle whose cosecant is the input value. The range for \( \operatorname{arccsc} \) is \( \left( 0, \frac{\pi}{2} \right] \cup \left( \pi, \frac{3\pi}{2} \right] \). First, we'll evaluate \( \csc \left( \frac{9 \pi}{8} \right) \).
02
Evaluate \( \csc \left( \frac{9 \pi}{8} \right) \)
To find \( \csc \left( \frac{9 \pi}{8} \right) \), we first recognize that \( \csc(\theta) = \frac{1}{\sin(\theta)} \). At \( \frac{9 \pi}{8} \), the angle is located in the third quadrant where sine is negative. Convert \( \frac{9 \pi}{8} \) into degrees or a more familiar angle if necessary. \( \sin \left( \frac{9 \pi}{8} \right) = -\sin \left( \frac{\pi}{8} \right) \). Thus, \( \csc \left( \frac{9 \pi}{8} \right) = - \csc \left( \frac{\pi}{8} \right) \).
03
Evaluate \( \operatorname{arccsc} \)
Since the function is \( \operatorname{arccsc}(\csc(\theta)) = \theta \), we need a \( \theta \) in the range \( \left( 0, \frac{\pi}{2} \right] \cup \left( \pi, \frac{3\pi}{2} \right] \) such that \( \csc(\theta) = -\csc \left( \frac{\pi}{8} \right) \). In the allowable range, the angle equivalent to \( \frac{9\pi}{8} \) with the same cosecant value in the range of arccsc is \( \theta = \frac{7\pi}{8} \).
04
Final Answer
Thus, the exact value of \( \operatorname{arccsc} \left( \csc \left( \frac{9 \pi}{8} \right) \right) \) is \( \frac{7 \pi}{8} \), since for \( \csc \theta = -\csc \left( \frac{\pi}{8} \right) \), \( \theta \) in the defined range is \( \frac{7\pi}{8} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Arcsecant
The arcsecant function, denoted as \( \operatorname{arcsec} \), is the inverse of the secant function. While it might not be as commonly encountered as sine or cosine inverses, it plays a crucial role when dealing with trigonometric equations involving secant values.
- The arcsecant function returns an angle \( \theta \) such that \( \sec(\theta) = x \), where \( x \) is the given input value.
- The principal range for arcsecant is typically \( [0, \frac{\pi}{2}) \cup [\pi, \frac{3\pi}{2}) \), which avoids the undefined area around the y-axis of the secant graph.
- This function is particularly useful when processing trigonometric identities or solving equations that involve secant values.
Arccosecant
Arccosecant, written as \( \operatorname{arccsc} \), serves as the inverse of the cosecant function. Like other inverse trigonometric functions, it helps to find angles from given trigonometric values.
- It returns the angle \( \theta \) corresponding to a given cosecant value such that \( \csc(\theta) = x \).
- The typical range for \( \operatorname{arccsc} \) is \( (0, \frac{\pi}{2}] \cup (\pi, \frac{3\pi}{2}] \). This is chosen to avoid the problematic discontinuities in the cosecant function.
- Understanding arccosecant is essential when handling problems where the direct calculation of angles from cosecant values is needed.
Cosecant Function
The cosecant function, represented as \( \csc(\theta) \), is one of the six fundamental trigonometric functions. It is the reciprocal of the sine function.
- Defined as \( \csc(\theta) = \frac{1}{\sin(\theta)} \), it is undefined whenever \( \sin(\theta) = 0 \).
- As with other reciprocal functions, it exhibits asymptotic behavior where the sine function approaches zero.
- Cosecant is significant in various applications, including physics, engineering, and geometry.
Third Quadrant
In trigonometry, angles are often discussed according to their position in one of the four quadrants of the Cartesian plane. The third quadrant is where angles range from \( \pi \) to \( \frac{3\pi}{2} \).
- In this quadrant, the sine and cosine functions both yield negative values.
- This has implications for their reciprocal functions: cosecant and secant, which will also be negative in this region.
- Understanding the behavior of angles and functions in the third quadrant is crucial for accurately evaluating trigonometric functions and solving equations.
Trigonometric Identities
Trigonometric identities are equations that involve trigonometric functions, holding true for all legitimate angles along their domains. They play a vital role in simplifying expressions and solving equations.
- One common identity is the Pythagorean identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \).
- Reciprocal identities, such as \( \csc(\theta) = \frac{1}{\sin(\theta)} \), are essential for transforming expressions into their simplified forms.
- There are also co-function identities, which relate functions at complementary angles, further aiding in the solution of complex trigonometric equations.
