Question

Please help me find all solutions to the equation below on the interval [0,2pi), thank you!

Answer 1

`x=(5\pi)/(16),(13\pi)/(16),(21\pi)/(16),(29\pi)/(16),(7\pi)/(16),(15\pi)/(16),(23\pi)/(16),(31\pi)/(16)`

Explanation:

We have the following equation:

`\sin \: 4x=-(√(2))/(2)`

We solve for x:

`\begin{gathered} 4x=\arcsin \mleft(-(√(2))/(2)\mright) \\ 4x=(5\pi)/(4)+2\pi n,(7\pi)/(4)+2\pi n \\ x=(5\pi)/(16)+(1)/(2)\pi n,(7\pi)/(16)+(1)/(2)\pi n \\ x_1=(5\pi)/(16) \\ x_2=(5\pi)/(16)+(\pi)/(2)=(13\pi)/(16) \\ x_3=(5\pi)/(16)+2\cdot(\pi)/(2)=(21\pi)/(16) \\ x_4=(5\pi)/(16)+3\cdot(\pi)/(2)=(29\pi)/(16) \\ x_5=(7\pi)/(16) \\ x_6=(7\pi)/(16)+(\pi)/(2)=(15\pi)/(16) \\ x_7=(7\pi)/(16)+2\cdot(\pi)/(2)=(23\pi)/(16) \\ x_8=(7\pi)/(16)+3\cdot(\pi)/(2)=(31\pi)/(16) \end{gathered}`