Question

Is `cos 27^@ + cos 93^@ + cos 143^@ = 0` ?

Answer 1

No

Explanation:

As a quick check, note that the trigonometric functions of any angle that is a multiple of `3^@` is expressible using square roots, but the trigonometric functions of any other multiple of `1^@` are not.

See https://socratic.org/s/aMANhumV for example.

Note that `27^@` and `93^@` are both multiples of `3`, so `cos 27^@` and `cos 93^@` are expressible in terms of square roots.

Note that `143^@` is a whole number of degrees but not a multiple of `3^@`. So `cos 143^@` is not expressible in terms of square roots.

Therefore it is not possible that `cos 27^@ + cos 93^@ + cos 143^@ = 0`, since otherwise we would have:

`cos 143^@ = -cos 27^@-cos 93^@`

with the left hand expression not being expressible in terms of square roots, while the right hand expression is.

In fact, we find:

`cos 27^@ + cos 93^@ + cos 147^@ = 0`