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Chapter 5: Problem 34

Is arcsine an even function, an odd function, or neither?

Short Answer

Expert verified
The arcsine function (\(\arcsin(x)\)) is an odd function.

Step by step solution

01

Test if \(\arcsin(-x) = \arcsin(x)\)

Let's start by evaluating \(\arcsin(-x)\). We know sine function is periodic with period \(-2n\pi\), and \(\sin(x) = \sin(x + 2n\pi)\). Consider the case when \(\sin(x) = -x\), then it means there exists a value \(x\), such that \(\sin(\arcsin(-x) +2n\pi) = -x\). If \(\sin(\arcsin(-x) + 2n\pi) = -x\), then \(\sin(\arcsin(-x)) = \sin(\arcsin(-x) + 2n\pi)\). However, this condition does not hold, as the sine function does not remain constant with the addition of multiples of its period. It alternates signs based on the quadrant of the periodic function. Hence, this doesn't satisfy the requirement for arcsine to be an even function. #Step 2: Test if arcsine is odd#
02

Test if \(\arcsin(-x) = -\arcsin(x)\)

Let's evaluate \(\arcsin(-x)\) again, but this time using the inverse relationship with sine function. We know \(\sin(\arcsin(x)) = x\), then when we multiply the argument \(-x\) inside the sine function with a negative sign like, \(-\sin(\arcsin(x)) = -x\), this original relationship still holds. We have \(\sin(-\arcsin(x)) = -x\), and applying the inverse function, we get \(\arcsin(-x) = -\arcsin(x)\), which is the condition for an odd function. As the arcsine function satisfies the condition for being an odd function, we can conclude: The arcsine function (\(\arcsin(x)\)) is an odd function.

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Most popular questions from this chapter

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