Question

Math:

Solve the equation. (Enter your answers as a comma-separated list. Use n as an integer constant. Enter your response in radians.)



`√(3) \csc x-2 = 0`


and


Solve the equation.


cos x + 1 = −cos x


x =

Answer 1

a)
`x_(1) = (\pi)/(3)\pm 2\pi \cdot i`,
`\forall i \in \mathbb{N}_(O)`,
`x_(2) = (5\pi)/(6)\pm 2\pi\cdot i`,
`\forall i \in \mathbb{N}_(O)`, b)
`x_(1) = (\pi)/(3)\pm 2\pi \cdot i`,
`\forall i \in \mathbb{N}_(O)`,
`x_(2) = (5\pi)/(3)\pm 2\pi\cdot i`,
`\forall i \in \mathbb{N}_(O)`

Explanation:

a) The equation must be rearranged into a form with one fundamental trigonometric function first:

`√(3)\cdot \csc x - 2 = 0`

`√(3) \cdot \left((1)/(\sin x) \right) - 2 = 0`

`√(3) - 2\cdot \sin x = 0`

`\sin x = (√(3))/(2)`

`x = \sin^(-1) (√(3))/(2)`

Value of x is contained in the following sets of solutions:

`x_(1) = (\pi)/(3)\pm 2\pi \cdot i`,
`\forall i \in \mathbb{N}_(O)`

`x_(2) = (5\pi)/(6)\pm 2\pi\cdot i`,
`\forall i \in \mathbb{N}_(O)`

b) The equation must be simplified first:

`\cos x + 1 = - \cos x`

`2\cdot \cos x = -1`

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

`x = \cos^(-1) \left(-(1)/(2) \right)`

Value of x is contained in the following sets of solutions:

`x_(1) = (\pi)/(3)\pm 2\pi \cdot i`,
`\forall i \in \mathbb{N}_(O)`

`x_(2) = (5\pi)/(3)\pm 2\pi\cdot i`,
`\forall i \in \mathbb{N}_(O)`