Problem 4. The number zero is written on the board. Peter is allowed to perform the following operations:

- apply a trigonometric (sin, $\cos$, $\operatorname{tg}$, or ctg) or inverse trigonometric (arcsin, arccos, $\operatorname{arctg}$, or arcctg) function to one of the numbers written on the board and write the result on the board;
- write the quotient or product of two already written numbers on the board.

Help Peter write $\sqrt{3}$ on the board.
Solution. Petr can, for example, perform the following calculations:

- $\cos 0=1$
- $\arctan 1=\frac{\pi}{4}$
- $\arccos 0=\frac{\pi}{2} ;$
- $\frac{\pi}{4}: \frac{\pi}{2}=\frac{1}{2}$
- $\arccos \frac{1}{2}=\frac{\pi}{3}$
- $\operatorname{tg} \frac{\pi}{3}=\sqrt{3}$.

Remark. Many sequences of operations can lead to the required result.

## Criteria

1 p. The number $\frac{\pi}{2}$ or $\frac{\pi}{4}$ is obtained.

2 6. The numbers $\frac{\pi}{2}$ and $\frac{\pi}{4}$ are obtained.

2 6. The number $\frac{1}{2}$ is obtained.

3 p. The number $\frac{\pi}{3}$ or $\frac{\pi}{6}$ is obtained.

4 p. Correct solution.