Question

What is `cos 2^@` ?

Answer 1

`cos 2^@ ~~ 0.99939083`

Explanation:

`cos 2^@ ~~ 0.99939083` is an irrational number which is not expressible in terms of real `n`th roots.

Note that the trigonometric functions of any integer multiple of `3^@` are expressible in terms of radicals, but other whole number of degrees do not have any such expressions.

We can effectively calculate approximations for the `sin` or `cos` of small angles using their series, once we have converted the angles to radians.

So:

`sin x = x/(1!)-x^3/(3!)+x^5/(5!)-x^7/(7!)+...`

`cos x = 1/(0!)-x^2/(2!)+x^4/(4!)-x^6/(6!)+...`

We can also use the accurate approximation:

`pi ~~ 355/113`

So:

`2^@ = 2/180*pi ~~ 355/(90*113) = 71/2034` radians

Then:

`cos 2^@ ~~ cos (71/2034)`

`color(white)(cos 2^@) ~~ 1-71^2/(2*2034^2)+71^4/(24*2034^4)`

`color(white)(cos 2^@) ~~ 0.99939083`