Solve the equation for exact solutions over the interval 0,2π
(cscx + 2)(cscx - squareroot2) = 0
Hello, nee!
The problem is 90% solved for you already.
Exactly where is your difficulty?
It's already factored!Solve the equation for exact solutions over the interval $\displaystyle [0,2\pi]$
$\displaystyle (\csc x + 2)(\csc x - \sqrt{2}) \:= \:0$
Set each factor equal to zero and solve . . .
$\displaystyle \csc x + 2 \:=\:0 \quad\Rightarrow\quad \csc x \:=\:-2\quad\Rightarrow\quad x \:=\:\frac{7\pi}{6},\;\frac{11\pi}{6}$
$\displaystyle \csc x - \sqrt{2}\:=\:0 \quad\Rightarrow\quad \csc x \:=\:\sqrt{2}\quad\Rightarrow\quad x \:=\:\frac{\pi}{4},\;\frac{3\pi}{4}$
set each factor equal to zero. solutions are right off the unit circle, which you have memorized, right?
$\displaystyle \csc{x} = -2$
$\displaystyle \sin{x} = -\frac{1}{2}$
$\displaystyle x = \frac{7\pi}{6}$ , $\displaystyle x = \frac{11\pi}{6}$
$\displaystyle \csc{x} = \sqrt{2}$
$\displaystyle \sin{x} = \frac{1}{\sqrt{2}}$
$\displaystyle x = \frac{\pi}{4}$ , $\displaystyle x = \frac{3\pi}{4}$
Soroban getting too fast in my old age.