Final answer to the problem
$x=\frac{1}{4}\pi+2\pi n,\:x=\frac{3}{4}\pi+2\pi n\:,\:\:n\in\Z$
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Step-by-step Solution
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- Express in terms of sine and cosine
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1
Calculate the square root of $2$
$\sin\left(x\right)+\frac{-\sqrt[4]{4}}{2}=0$
2
Multiply $-1$ times $\sqrt[4]{4}$
$\sin\left(x\right)-\frac{\sqrt[4]{4}}{2}=0$
3
Divide $-\sqrt[4]{4}$ by $2$
$\sin\left(x\right)-\sqrt[4]{\frac{1}{4}}=0$
4
We need to isolate the dependent variable $x$, we can do that by simultaneously subtracting $-\sqrt[4]{\frac{1}{4}}$ from both sides of the equation
$\sin\left(x\right)=\sqrt[4]{\frac{1}{4}}$
5
The angles where the function $\sin\left(x\right)$ is $\sqrt[4]{\frac{1}{4}}$ are
$x=45^{\circ}+360^{\circ}n,\:x=135^{\circ}+360^{\circ}n$
6
The angles expressed in radians in the same order are equal to
$x=\frac{1}{4}\pi+2\pi n,\:x=\frac{3}{4}\pi+2\pi n\:,\:\:n\in\Z$
Final answer to the problem
$x=\frac{1}{4}\pi+2\pi n,\:x=\frac{3}{4}\pi+2\pi n\:,\:\:n\in\Z$