Question

If $\cos \left(x\right)+\sqrt{\sin \left(x\right)}=0$ then $x$

Answer 1

Find the value of given expression

Given, $\cos \left(x\right)+\sqrt{\sin \left(x\right)}=0$

$$\begin{gathered} \Rightarrow \cos \left(x\right)=-\sqrt{\sin \left(x\right)} \\ \Rightarrow {\left(\cos \left(x\right)\right)}^2={\left(-\sqrt{\sin \left(x\right)}\right)}^2 \\ \Rightarrow {\cos }^2\left(x\right)=\sin \left(x\right) \\ \Rightarrow 1-{\sin }^2\left(x\right)=\sin \left(x\right);\left[\because {\sin }^2\left(x\right)+{\cos }^2\left(x\right)=1\right] \\ \Rightarrow {\sin }^2\left(x\right)+\sin \left(x\right)-1=0 \\ \Rightarrow \sin \left(x\right)=\frac{-1\pm \sqrt{1^2-4·1·\left(-1\right)}}{2} \\ \Rightarrow \sin \left(x\right)=\frac{-1\pm \sqrt{5}}{2} \\ \Rightarrow \sin \left(x\right)=\frac{-1+\sqrt{5}}{2};\left[\sin \left(x\right)\ne \frac{-1-\sqrt{5}}{2}\because -1\le \sin \left(x\right)\le 1\right] \\ \Rightarrow x={\sin }^{-1}\left(\frac{\sqrt{5}-1}{2}\right) \end{gathered}$$

Hence, the value of $x$ is ${\sin }^{-1}\left(\frac{\sqrt{5}-1}{2}\right)$.