Is the function $f(x) = cos x$ even, odd or neither?

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Is the function $f(x) = cos x$ even, odd or neither?

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Answer 1 · 2015-10-31T11:09:44

Even

Explanation:

$cos(x)=cos(-x)$ , therefore cosine is an even function.

Read more:
http://mathworld.wolfram.com/Cosine.html

Answer 2 · 2018-01-08T20:27:29

To prove that $cos(theta)$ is even, i.e. that $cos(-theta)=cos(theta)$ , we can use the unit circle, which mind you, is the definition of cosine arguments outside the interval $[0,pi/2]$ .

The unit circle is a circle of radius one centered at the origin. We can draw the following constructions for $cos(theta)$ and $cos(-theta)$ :
i.imgur.com
We see that the points $(cos(theta),sin(theta))$ and $(cos(-theta),sin(-theta))$ are on the same vertical line. Since the unit circle is in a cartesian coordinate system, this must mean they have the same $x$ -coordinates.

The first point has an $x$ -coordinate of $cos(theta)$ , and the second has an $x$ -coordinate of $cos(-theta)$ , and they must be equal, so it quite easily follows that:
$cos(-theta)=cos(theta)$

Which proves that cosine is an even function.

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Is the function $f(x) = cos x$ even, odd or neither?

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Explanation:

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