Even Symmetry Of Cosine Function And Deriving Its Result For Cos(-X) = Cos(X)



If we have a real number x, then -x represents the additive inverse of x which means that it is the number that when added to x gives zero.

Now consider the function cos(x), which is the value of the cosine of the angle x (in radians). The angle x can be any real number.

Therefore, cos(-x) represents the value of the cosine of the angle -x. The cosine function is an even function which means that it has symmetry around the y-axis. In other words, cos(-x) = cos(x) for all values of x.

To show this, we can use the trigonometric identity:

cos(-x) = cos(x + π)

where π is the mathematical constant pi (approximately 3.14159).

Since π is an odd number of multiples of 180 degrees, adding π to an angle is equivalent to reflecting it across the x-axis. Therefore, cos(x + π) is the same as cos(x) but with the opposite sign, since the cosine function is negative in the second and third quadrants.

Thus, we have:

cos(-x) = cos(x + π) = -cos(x)

But since cos(-x) = cos(x) by the even symmetry of the cosine function, we can put these together to get:

cos(x) = -cos(x)

Solving for cos(x), we get:

2 cos(x) = 0

cos(x) = 0

Therefore, we have determined that cos(-x) = cos(x) = 0 whenever x is an odd multiple of π/2 or equivalently, x = (2n + 1)π/2, where n is an integer.

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