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Identities without variables

The curious identity

is a special case of an identity that contains one variable:

Similarly:

The same cosine identity in radians is

Similarly:

The following is perhaps not as readily generalized to an identity containing variables (but see explanation below):

Degree measure ceases to be more felicitous than radian measure when we consider this identity with 21 in the denominators:

The factors 1, 2, 4, 5, 8, 10 may start to make the pattern clear: they are those integers less than 21/2 that are relatively prime to (or have no prime factors in common with) 21. The last several examples are corollaries of a basic fact about the irreducible cyclotomic polynomials: the cosines are the real parts of the zeroes of those polynomials; the sum of the zeroes is the Möbius function
evaluated at (in the very last case above) 21; only half of the zeroes
are present above. The two identities preceding this last one arise in
the same fashion with 21 replaced by 10 and 15, respectively.

Many of those curious identities stem from more general facts like the following:

^{}

and

Combining these gives us

If

*n* is an odd number (

*n* = 2

*m* + 1) we can make use of the symmetries to get

The transfer function of the Butterworth low pass filter
can be expressed in terms of polynomial and poles. By setting the
frequency as the cutoff frequency, the following identity can be proved:

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Computing π

An efficient way to compute π is based on the following identity without variables, due to Machin:

or, alternatively, by using an identity of Leonhard Euler:

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A useful mnemonic for certain values of sines and cosines

For certain simple angles, the sines and cosines take the form

for 0 ≤

*n* ≤ 4, which makes them easy to remember.

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Miscellany

With the golden ratio φ:

Also see exact trigonometric constants.

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An identity of Euclid

Euclid showed in Book XIII, Proposition 10 of his

*Elements*
that the area of the square on the side of a regular pentagon inscribed
in a circle is equal to the sum of the areas of the squares on the
sides of the regular hexagon and the regular decagon inscribed in the
same circle. In the language of modern trigonometry, this says:

Ptolemy used this proposition to compute some angles in his table of chords.

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