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Chapter 5: Problem 33

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

Short Answer

Expert verified
The arccosine function is neither an even function nor an odd function, as it does not satisfy the properties for either evenness or oddness. Specifically, arccos(-x) = 蟺 - arccos(x), which doesn't conform to either the even or odd function properties.

Step by step solution

01

Identifying the domain of the arccosine function

The domain of the arccosine function, denoted as arccos(x) or cos^(-1)(x), is the set of all values x for which the function is defined. Since the cosine function (cos(胃)) ranges from -1 to 1, the domain of the arccosine function is given by the interval [-1, 1].
02

Even function test

We will now test whether the arccosine function is even. To do this, we need to find out if the property f(x) = f(-x) holds true for all x in the domain [-1, 1]. Let's take the arccos(x) function and find its value when we replace x with -x: arccos(-x) = 胃 <=> cos(胃) = -x. The angle 胃 must satisfy the condition: cos(胃) = -x. Now, let's consider the arccos(x) function, where cos(胃) = x. Notice that for cos(胃) = x, we have the following relationship: cos(蟺 - 胃) = -cos(胃) = -x. So, when -x is replaced with arccos(-x), we get: arccos(-x) = 蟺 - arccos(x). Since arccos(-x) 鈮� arccos(x), the arccosine function does not satisfy the even function property and, therefore, is not an even function.
03

Odd function test

Now, we will test whether the arccosine function is odd. To do this, we need to find out if the property f(x) = -f(-x) holds true for all x in the domain [-1, 1]. We notice that in this case, f(x) = arccos(x), and -f(-x) = -arccos(-x). From Step 2, we already found that arccos(-x) = 蟺 - arccos(x). By replacing arccos(-x) with 蟺 - arccos(x), we get: -arccos(-x) = - (蟺 - arccos(x)) = arccos(x) - 蟺. Since arccos(x) 鈮� arccos(x) - 蟺, the arccosine function does not satisfy the odd function property and, therefore, is not an odd function.
04

Conclusion

As the arccosine function satisfies neither the even function property nor the odd function property, we conclude that it is neither an even function nor an odd function.

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Show that $$(\cos x+\sin x)^{2}=1+\sin (2 x)$$ for every number \(x\).

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