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Magazine Chapter 10: Problem 36
Find the exact value. \(\operatorname{arccsc}(\sqrt{2})\)
Short Answer
Expert verified
The exact value is \(\frac{\pi}{4}\).
Step by step solution
01
Understand the Function
The function \(\operatorname{arccsc}\) is the inverse of the cosecant function \(\operatorname{csc}\). It returns the angle whose cosecant is the given value. In this problem, we need to find the angle \(\theta\) such that \(\operatorname{csc}(\theta) = \sqrt{2}\).
02
Relate Cosecant to Sine
Recall the identity \(\operatorname{csc}(\theta) = \frac{1}{\sin(\theta)}\). If \((\operatorname{csc}(\theta) = \sqrt{2})\), then \((\sin(\theta) = \frac{1}{\sqrt{2}})\).
03
Recall the Known Values of Sine
Identify the angle \((\theta)\) on the unit circle that corresponds to \((\sin(\theta) = \frac{1}{\sqrt{2}})\). The angle that satisfies this is \((\theta = \frac{\pi}{4})\) because \((\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2})\), and \((\sin(\theta) = \frac{1}{\sqrt{2}})\) is equivalent to that when rationalized.
04
Check for Additional Solutions
Consider the cosecant function's symmetry. Since \((\sin(\pi - \theta) = \sin(\theta))\), \(\theta = \frac{\pi}{4}\) mirrored over the y-axis also satisfies \(\sin(\theta) = \frac{1}{\sqrt{2}}\). However, \(\operatorname{arccsc}\) typically returns the principal value, which is the smallest positive angle, \(\theta = \frac{\pi}{4}\).
05
Conclusion
Having identified the principle angle, we conclude that \((\operatorname{arccsc}(\sqrt{2}) = \frac{\pi}{4})\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Arccsc Function
The **Arccsc (arc cosecant)** function is the inverse of the cosecant function. It is denoted as \( \operatorname{arccsc} \). This function helps us find the angle whose cosecant equals a specific value. For example, if you want to find \( \theta \) such that \( \operatorname{csc}(\theta) = \sqrt{2} \), you use the arccsc function.When dealing with inverse trigonometric functions like arccsc, we're looking for angles rather than the trigonometric ratios themselves. Remember, since the cosecant function \( \operatorname{csc}(\theta) \) is defined as the reciprocal of the sine \( \sin(\theta) \), you can transform an arccsc problem into a sine problem.
Unit Circle
The **Unit Circle** is a fundamental concept in trigonometry. It is a circle with a radius of 1, centered at the origin of a coordinate plane. This circle helps simplify the understanding of trigonometric functions and their inverses.Important attributes of the unit circle:
- The coordinates on the unit circle correspond to the cosine and sine of angles \( (\cos(\theta), \sin(\theta)) \).
- Angles are usually measured in radians. For instance, \( \theta = \frac{\pi}{4} \) corresponds to both \( \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \) and \( \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \).
- It provides a clear reference for determining trigonometric function values and their principal angles over a 360-degree or \( 2\pi \) span.
Cosecant Function
The **Cosecant Function**, denoted as \( \operatorname{csc}(\theta) \), is the reciprocal of the sine function. It represents the ratio between the hypotenuse and the opposite side in a right triangle: \( \operatorname{csc}(\theta) = \frac{1}{\sin(\theta)} \).Key characteristics of the cosecant function include:
- It is undefined whenever \( \sin(\theta) = 0 \) because it would imply division by zero.
- The function tends to infinity as \( \sin(\theta) \) approaches zero, reflecting its reciprocal nature.
- In the unit circle context, high values of the cosecant function occur at angles where sine is close to zero because \( \operatorname{csc}(\theta) \) is its reciprocal.
Principal Value
In trigonometry, the **Principal Value** refers to the main value of an inverse trigonometric function. It ensures unique and specific results when multiple angles could satisfy the condition.Understanding principal values:
- For \( \operatorname{arccsc} \), principal values are generally limited to \([ -\frac{\pi}{2}, \frac{\pi}{2} ]\), excluding zero.
- This range ensures a one-to-one correspondence with principal values, preventing ambiguity.
- For our example \( \operatorname{arccsc}(\sqrt{2}) \), the principal value is \( \theta = \frac{\pi}{4} \), as it is the smallest positive angle fulfilling the given cosecant condition.
