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Chapter 10: Problem 126

Assume that the range of arcsecant is \(\left[0, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \pi\right]\) and that the range of arccosecant is \(\left[-\frac{\pi}{2}, 0\right) \cup\left(0, \frac{\pi}{2}\right]\) when finding the exact value. \(\operatorname{arccsc}\left(\csc \left(\frac{2 \pi}{3}\right)\right)\)

Short Answer

Expert verified
\(\frac{\pi}{3}\) is the exact value.

Step by step solution

01

Understand the Problem

To solve the problem, we need to find the exact value of the expression \(\operatorname{arccsc}\left(\csc \left(\frac{2 \pi}{3}\right)\right)\). We'll start by understanding what each part of the expression represents.
02

Evaluate the Cosecant

The cosecant function is the reciprocal of the sine function. Thus, \(\csc \left(\frac{2 \pi}{3}\right) = \frac{1}{\sin \left(\frac{2 \pi}{3}\right)}\). The sine of \(\frac{2 \pi}{3}\) is \(\sin \left(\frac{2 \pi}{3}\right) = \sin \left(\pi - \frac{\pi}{3}\right) = \sin \left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\). So, \(\csc \left(\frac{2 \pi}{3}\right) = \frac{2}{\sqrt{3}}\). To simplify, multiply by \(\frac{\sqrt{3}}{\sqrt{3}}\), resulting in \(\frac{2\sqrt{3}}{3}\).
03

Apply the Arccosecant Function

The arccosecant function returns the angle whose cosecant is the given value. We need \( y = \operatorname{arccsc} \left( \frac{2\sqrt{3}}{3} \right)\) such that \( \csc(y) = \frac{2\sqrt{3}}{3}\). Since \( y = \arcsin\left(\frac{\sqrt{3}}{2}\right)\), the angle \(y\) must be in the range \(\left[-\frac{\pi}{2}, 0\right) \cup \left(0, \frac{\pi}{2}\right]\). The angle for this sine value is \(\frac{\pi}{3}\) since it lies in \(\left(0, \frac{\pi}{2}\right]\).
04

Conclusion

Thus, the exact value of \(\operatorname{arccsc}\left(\csc \left(\frac{2 \pi}{3}\right)\right)\) is \(\frac{\pi}{3}\) as it satisfies the arccosecant conditions and the sine value simplifies to a familiar angle in the specified range.

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Key Concepts

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

Arcsecant
The arcsecant function, often written as \( \text{arcsec}(x) \), is the inverse of the secant function. In mathematical terms, it helps us find the angle whose secant is \( x \). The secant function itself is defined as the reciprocal of the cosine, that is, \( \sec(\theta) = \frac{1}{\cos(\theta)} \). Hence, to understand arcsecant, we should know:
  • It is related to the cosine function because secant (\( \sec \)) is the reciprocal of cosine (\( \cos \)).
  • The range of arcsecant is typically restricted to ensure it is a function, often \( [0, \frac{\pi}{2}) \cup (\frac{\pi}{2}, \pi] \).
  • Arcsecant is used to solve equations involving secant in various fields such as physics and engineering, where angles need to be determined from ratios of sides.
Knowing these points about arcsecant can deepen understanding of how we use it in trigonometric identities and when converting angles to different measures.
Arccosecant
The arccosecant function, written as \( \text{arccsc}(x) \), is the inverse of the cosecant function. It allows us to find the angle whose cosecant is \( x \). Cosecant is the reciprocal of the sine function, given by \( \csc(\theta) = \frac{1}{\sin(\theta)} \). Here鈥檚 what you should know when handling arccosecant:
  • The range is restricted to \( \left[-\frac{\pi}{2}, 0\right) \cup \left(0, \frac{\pi}{2}\right] \) to ensure it is a true function.
  • Arccosecant is frequently utilized in trigonometry to revert from trigonometric values back to angles, critical in calculus and physics problems.
  • It helps in understanding relationships between angles and their trigonometric ratios, especially when translating between arcsin, arctan, and arcsec.
Understanding arccosecant is essential for solving problems involving oscillations or waves, where angles define important physical properties.
Cosecant
The cosecant function, denoted as \( \csc(\theta) \), is the reciprocal of the sine function. This means that if \( \sin(\theta) \) denotes the sine of an angle, then \( \csc(\theta) = \frac{1}{\sin(\theta)} \). Here are some key aspects of cosecant:
  • The function is undefined at \( \theta = n\pi \), where \( n \) is an integer, because \( \sin(\theta) = 0 \) at these points, leading to division by zero.
  • Cosecant is used to calculate the lengths of sides in right-angled triangles when the angle and hypotenuse are known.
  • In terms of graphs, cosecant has vertical asymptotes at \( \theta = n\pi \) and displays a wave-like pattern similar to sine but flipped and stretched.
Cosecant allows us to explore and understand how angles relate to arc lengths and sector areas, particularly useful in astronomy and engineering.
Sine Function
The sine function is among the most fundamental trigonometric functions, represented as \( \sin(\theta) \). It expresses the ratio of the length of the side opposite the angle to the hypotenuse in a right-angled triangle. Key characteristics about the sine function include:
  • The function is periodic with a period of \( 2\pi \), meaning that the values repeat every \( 2\pi \) units.
  • The range of sine is \([-1, 1]\), capturing the maximum and minimum values that \( \sin(\theta) \) can take.
  • Sine is critical in modeling periodic phenomena such as sound waves, light waves, and tides, making it central in physics and engineering.
A deep understanding of the sine function forms the foundation for learning about other trigonometric functions, such as cosine and tangent, and their applications in various scientific fields.

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Most popular questions from this chapter

Rewrite the quantity as algebraic expressions of \(x\) and state the domain on which the equivalence is valid. $$ \sin \left(\frac{1}{2} \arctan (x)\right) $$

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