Question

Solve for the exact solutions in the interval

List your answers separated by a comma, if it has no real solutions, enter DNE.

Answer 1

`√(2) \sin(3x) - 1 = 0`

`√(2) \sin(3x) = 1`

`\sin(3x) = ( √(2) )/(2)`

3x = π/4 + 2kπ or 3x = 3π/4 + 2kπ

x = π/12 + 2kπ/3 or x = π/4 + 2kπ/3

0 < π/12 + 2kπ/3 < 2π

0 < 1/12 + 2k/3 < 2

0 < 1 + 8k < 24

-1 < 8k < 23, so k = 0, 1, 2

x = π/12, 3π/4, 17π/12

0 < π/4 + 2kπ/3 < 2π

0 < 1/4 + 2k/3 < 2

0 < 3 + 8k < 24

-3 < 8k < 21, so k = 0, 1, 2

x = π/4, 11π/12, 19π/12