Answer
The solution on the interval $[0,2\pi )$, correct to four decimal places is $0.4636$ , $2.6780$, $3.6052$ or $5.8195$.
Work Step by Step
Let us consider the provided expression:
$5{{\sin }^{2}}x-1=0$
Add 1 on both sides:
$\begin{align}
& 5{{\sin }^{2}}x-1+1=0+1 \\
& 5{{\sin }^{2}}x=1
\end{align}$
And divide by 5 on both sides:
$\begin{align}
& \frac{5{{\sin }^{2}}x}{5}=\frac{1}{5} \\
& {{\sin }^{2}}x=\frac{1}{5}
\end{align}$
And put the square root on both sides:
$\begin{align}
& \sin x=\pm \sqrt{\frac{1}{5}} \\
& \approx \pm 0.4472
\end{align}$
Since, the value of $\sin x\approx +0.4472$
$x\approx {{\sin }^{-1}}\left( +0.4472 \right)$
And the value of $x$ from $\sin x\approx -0.4472$ is:
$x\approx {{\sin }^{-1}}\left( -0.4472 \right)$
We have to calculate the equation with the help of a calculator in radian mode:
$\theta ={{\sin }^{-1}}\left( +0.4472 \right)\approx 0.4636$
$\theta ={{\sin }^{-1}}\left( -0.4472 \right)\approx -0.4636$
So,
$x\approx 0.4636$
Also, the sine value is positive in the II quadrant and the value of $\pi $ is $3.14159$. Therefore,
$\begin{align}
& x=\pi -0.4636 \\
& =3.14159-0.4636 \\
& \approx 2.6780
\end{align}$
The sine value is negative in the III and IV quadrants and the value of $\pi $ is $3.14159$.
So,
$\begin{align}
& x=\pi +0.4636 \\
& =3.14159+0.4636 \\
& \approx 3.6052
\end{align}$
Or,
$\begin{align}
& x=2\pi -0.4636 \\
& =2\left( 3.14159 \right)-0.4636 \\
& =6.28318-0.4636 \\
& \approx 5.8195
\end{align}$
Thus, the solution on the interval $[0,2\pi )$, correct to four decimal places is $0.4636$ , $2.6780$ , $3.6052$ or $5.8195$.
