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Magazine Chapter 10: Problem 46
In Exercises \(41-48\), 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. $$ \operatorname{arccsc}(-\sqrt{2}) $$
Short Answer
Expert verified
\( \operatorname{arccsc}(-\sqrt{2}) = \frac{5\pi}{4} \).
Step by step solution
01
Understanding the Problem
We need to find the exact value of \( \operatorname{arccsc}(-\sqrt{2}) \). Recall that \( \operatorname{arccsc}(x) \) is the angle \( \theta \) such that \( \csc(\theta) = x \). The range for \( \operatorname{arccsc} \) is \( \left(0, \frac{\pi}{2}\right] \cup \left(\pi, \frac{3\pi}{2}\right] \).
02
Determine Reciprocal Value
Since \( \csc(\theta) = x \), we know that \( \sin(\theta) = \frac{1}{x} \). Therefore, if \( \csc(\theta) = -\sqrt{2} \), then \( \sin(\theta) = -\frac{1}{\sqrt{2}} = -\frac{\sqrt{2}}{2} \).
03
Identify the Possible Angles
The value \( -\frac{\sqrt{2}}{2} \) corresponds to \( \sin(\theta) \) for angles where the sine function is negative. Within our range, \( \sin(\theta) = -\frac{\sqrt{2}}{2} \) occurs at \( \theta = \frac{5\pi}{4} \) and \( \theta = \frac{7\pi}{4} \).
04
Select the Correct Angle from the Range
According to the range \( \left(0, \frac{\pi}{2}\right] \cup \left(\pi, \frac{3\pi}{2}\right] \), only \( \theta = \frac{5\pi}{4} \) falls within the valid intervals \( \left(\pi, \frac{3\pi}{2}\right] \).
05
Conclude the Solution
The angle \( \theta = \frac{5\pi}{4} \) is the correct value for \( \operatorname{arccsc}(-\sqrt{2}) \) as it satisfies the condition and falls within the specified range.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Arcsecant Range
The range of the inverse secant function, or arcsecant, is an essential component while working with trigonometric inverses. For the function arcsecant, the values it can return are confined to specific intervals. In this case, the range is between \( \left[0, \frac{\pi}{2}\right) \) and \( \left[\pi, \frac{3 \pi}{2}\right) \). This means if you take the arcsecant of any value, the angle that corresponds to that function will always be between 0 to \(\frac{\pi}{2} \) and \( \pi \) to \(\frac{3\pi}{2}\).
Here are some things to remember about arcsecant:
Here are some things to remember about arcsecant:
- It is the inverse of the secant function.
- Its range is formatted in radians, which is a common measurement for angles in higher mathematics.
- This limited range ensures that the function remains one-to-one, making it easier to find a unique angle for a given secant value.
Arccosecant Range
The arccosecant or inverse cosecant function has a specified range which dictates the possible angle outputs. When working with arccosecant, it's essential to understand its range, as it influences the solution to the problems you're attempting to solve. The standard range for arccosecant is \( \left(0, \frac{\pi}{2}\right] \cup \left(\pi, \frac{3\pi}{2}\right] \).
This means the angles produced by the arccosecant function are always found in these intervals:
This means the angles produced by the arccosecant function are always found in these intervals:
- From just above 0 to \( \frac{\pi}{2} \).
- And from just above \( \pi \) to \( \frac{3\pi}{2} \).
Exact Values of Trigonometric Functions
Finding the exact values of trigonometric functions is a crucial skill in trigonometry and calculus. It involves identifying precise output values for specific trigonometric inputs. To do so correctly, a deep understanding of the angles associated with common trigonometric values like \( \sin \theta = \frac{\sqrt{2}}{2} \) or \( \cos \theta = \frac{\sqrt{3}}{2} \) becomes necessary.
Here's how you can determine exact values in practice:
Here's how you can determine exact values in practice:
- Memorize the standard trig values for common angles like \( 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \) and \(\frac{\pi}{2} \).
- Understand the symmetry of the unit circle and how it affects trig functions in different quadrants.
Unit Circle
The unit circle is a fundamental tool for visualizing and understanding the behavior of trigonometric functions. Defined as a circle with a radius of one unit, it serves as a reference for angles and their corresponding sine and cosine values.
Using the unit circle, each angle in radians corresponds to a point on the circle, the x-coordinate is the cosine of the angle, and the y-coordinate is the sine.
Using the unit circle, each angle in radians corresponds to a point on the circle, the x-coordinate is the cosine of the angle, and the y-coordinate is the sine.
- It is an essential concept as it links angles in standard position to trigonometric values.
- The circle's symmetry helps identify angle relationships, especially for inverses.
- You can visualize actions like rotations and find reference angles quickly.
Reciprocal Trigonometric Functions
Reciprocal trigonometric functions offer alternative mathematical perspectives by inverting the original functions. These functions include secant \(\sec \theta\), cosecant \(\csc \theta\), and cotangent \(\cot \theta\). Understanding them is vital to solve complex trigonometric equations, like finding the arccosecant of a given value.
- Secant is the reciprocal of cosine: \(\sec \theta = \frac{1}{\cos \theta}\).
- Cosecant is the reciprocal of sine: \(\csc \theta = \frac{1}{\sin \theta}\).
- Cotangent is the reciprocal of tangent: \(\cot \theta = \frac{1}{\tan \theta}\).
