Question

How to solve this trigonometric equation: cos(x) * cos(2x) * cos (3x) = 1? Thank you!

Answer 1

`x=n pi, n in ZZ`

Explanation:

Since `|cos theta| le 1` , the only way the product of three cosines can be one is if either

• all three are 1
• two of them are -1, and the third is equal to +1

The first possibility would mean that `x=2npi, n in ZZ`.

The second would happen when `cos x = -1` (this would automatically lead to `cos2x = +1` and `cos 3x=-1`), which corresponds to `x = (2n+1)pi, n in ZZ`.

Combining the rtwo together we get the solution set

`x=n pi, n in ZZ`

Note : in the second case we could not have taken `cos2x=-1` because that would have led to `cos x = 1/2(1+cos2x) = 0`