Question

Solve the equation in the interval [0,2π]. If there is more than one solution write them separated by commas. (cos(x))^2= 1/16x=

Answer 1

The solutions to the equation
`$\cos^2(x) = (1)/(16)$` in the interval
`$[0, 2\pi]$` are:

`$x \approx 1.318, 1.823, 4.460, 5.965$`

To solve the equation
`$\cos^2(x) = (1)/(16)$` in the interval
`$[0, 2\pi]$`, we can take the square root of both sides to get rid of the squared term.

`$√(\cos^2(x)) = \sqrt{(1)/(16)}$`

Since
`$\cos(x)$` is always positive or zero, we can simplify the equation to:

`$\cos(x) = \pm(1)/(4)$`

To find the solutions in the interval
`$[0, 2\pi]$`, we need to consider the values of
`$x$` for which
`$\cos(x)$` is equal to
`$\pm(1)/(4)$`.

Using the unit circle or a calculator, we can find the angles where
`$\cos(x)$` is equal to
`$(1)/(4)$` or
`$-(1)/(4)$`.

For
`$\cos(x) = (1)/(4)$`, the solutions in the interval
`$[0, 2\pi]$` are approximately:

`$x \approx 1.318, 5.965$`

For
`$\cos(x) = -(1)/(4)$`, the solutions in the interval
`$[0, 2\pi]$` are approximately:

`$x \approx 1.823, 4.460$`