Question

Find all solutions to the equation cosxcos(3x)-sinxsin(3x)=0 on the interval [0,2pi]

Answer 1

We can condense the left side as

`\cos(x) \cos(3x) - \sin(x) \sin(3x) = \cos(x + 3x) = \cos(4x)`

Then the equation reduces to

`\cos(4x) = 0`

`4x = \pm \cos^(-1)(0) + 2n\pi`

`4x = \pm \frac\pi2 + 2n\pi`

`x = \pm \frac\pi8 + \frac{n\pi}2`

where
`n` is any integer.

In the interval
`[0,2\pi]`, we get the solutions

`x \in \left\{\frac\pi8, \frac{3\pi}8, \frac{5\pi}8, \frac{7\pi}8, \frac{9\pi}8, \frac{11\pi}8, \frac{13\pi}8, \frac{15\pi}8\right\}`