Q:
# How Is Cos 4pi Evaluated?

**There are two ways to evaluate cos 4? that will both give the answer of 1.** The best ways to evaluate involve the periodicity of the cosine function and the trigonometric addition formula for cosine.

First, the cosine function has a period of 2?, meaning that cos x = cos (x + 2?). Using this formula, we know that cos 0 = cos 2? = cos 4?. Recalling that cos 0 = 1, we know that cos 4? = cos 0 = 1.

In cases of cos 2n? or sin 2n?, where n is a whole number, this is a simple process. However, this process does not always involve simple terms like this. This is when the trigonometric addition formula for cosine is helpful.

First, suppose that cos z needs to be evaluated, and z = x + y, where x, y and z are real numbers.

Then, cos(x + y) = (cos x)(cos y) -(sin x)(sin y)

Using 2? as both x and y to make z = 4?,

cos 4? = cos (2? + 2?) = (cos 2?)(cos 2?) - (sin 2?)(sin 2?)

Now, recalling that cos 2? = 1 and sin 2? = 0,

cos 4? = (1) (1) - (0) (0) = 1.

While this method involves much more calculation, it works for any angle.

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