Question

Solve on the interval [0, 2pi] 2 sec x+5 = 1

Answer 1

2pi/3 and 4pi/3

Explanation:

this is the answer according to apex

Answer 2

Move the 5 to the other side:

`2\sec(x)=1-5=-4`

Divide both sides by 2:

`\sec(x) = -2`

Recall the definition:

`\sec(x)=-2 \iff (1)/(\cos(x))=-2`

Invert both sides

`\cos(x) = -(1)/(2)`

This is true when

`x=\pm (\pi)/(3)`

If you need both angles to be in [0,2pi], you can recall

`\cos\left(-(\pi)/(3)\right) = \cos\left(-(\pi)/(3)+2\pi\right) = \cos\left((5\pi)/(3)\right)`

So, the solutions are

`x=(\pi)/(3),\quad x=(5\pi)/(3)`