Question

How do I solve `sec(3x)-sqrt2 = 0` algebraically?

Answer 1

`x= pi/12+2/3pin`
`x= (7pi)/2+2/3pin`
n is an element of all integers

Explanation:

Let `u` be `3x`:
`sec(u)-sqrt2=0`

`secu= sqrt2`

Apply reciprocal identity:
`1/cosu= sqrt2`

`cosu= 1/sqrt2= sqrt2/2`

`u= pi/4 +2pin`
`u= (7pi)/4+2pin`

Now replace `u` with `3x` and solve for x:

`3x=pi/4 +2pin`
`3x=(7pi)/4+2pin`

`x= pi/12+2/3pin`
`x= (7pi)/2+2/3pin`

Here's a graph:
graph{sec(3x)-sqrt2 [-10, 10, -5, 5]}