Task 6. Find all values that the expression

$$
4 \arcsin x - \arccos y
$$

can take under the condition $x^{2} + y^{2} = 1$.
Answer: $\left[-\frac{5 \pi}{2} ; \frac{3 \pi}{2}\right]$.

Solution. Note that $x^{2}+y^{2}=1$ if and only if there exists some $\varphi \in[0 ; 2 \pi]$ such that $x=\sin \varphi, y=\cos \varphi$. Then the expression from the condition takes the form

$$
4 \arcsin \sin \varphi-\arccos \cos \varphi
$$

Let's consider several cases:

- $\varphi \in\left[0 ; \frac{\pi}{2}\right]:$ then $\arcsin \sin \varphi=\varphi, \arccos \cos \varphi=\varphi$, and

$$
4 \arcsin \sin \varphi-\arccos \cos \varphi=4 \varphi-\varphi=3 \varphi
$$

thus, for $\varphi \in\left[0 ; \frac{\pi}{2}\right]$ the expression (*) takes all values from the interval $\left[0 ; \frac{3 \pi}{2}\right]$;

- $\varphi \in\left[\frac{\pi}{2} ; \pi\right]:$ then $\arcsin \sin \varphi=\pi-\varphi, \arccos \cos \varphi=\varphi$, and

$$
4 \arcsin \sin \varphi-\arccos \cos \varphi=4(\pi-\varphi)-\varphi=4 \pi-5 \varphi
$$

thus, for $\varphi \in\left[\frac{\pi}{2} ; \pi\right]$ the expression (*) takes all values from the interval $\left[-\pi ; \frac{3 \pi}{2}\right]$;

- $\varphi \in\left[\pi ; \frac{3 \pi}{2}\right]:$ then $\arcsin \sin \varphi=\pi-\varphi, \arccos \cos \varphi=2 \pi-\varphi$, and

$$
4 \arcsin \sin \varphi-\arccos \cos \varphi=4(\pi-\varphi)-(2 \pi-\varphi)=2 \pi-3 \varphi
$$

thus, for $\varphi \in\left[\pi ; \frac{3 \pi}{2}\right]$ the expression (*) takes all values from the interval $\left[-\frac{5 \pi}{2} ;-\pi\right]$;

- $\varphi \in\left[\frac{3 \pi}{2} ; 2 \pi\right]:$ then $\arcsin \sin \varphi=\varphi-2 \pi, \arccos \cos \varphi=2 \pi-\varphi$, and

$$
4 \arcsin \sin \varphi-\arccos \cos \varphi=4(\varphi-2 \pi)-(2 \pi-\varphi)=-10 \pi+5 \varphi ;
$$

thus, for $\varphi \in\left[\frac{3 \pi}{2} ; 2 \pi\right]$ the expression (*) takes all values from the interval $\left[-\frac{5 \pi}{2} ; 0\right]$. Summarizing all of the above, we get that the expression (*) for $\varphi \in[0 ; 2 \pi]$ takes all values from the interval $\left[-\frac{5 \pi}{2} ; \frac{3 \pi}{2}\right]$.