Question

state of the system at any given time is determined by a set of values of n parameters. Each parameter can take one of three values: 1, 0, -1. It is known that the parameter value is defined by formula A1=Sin(A2 * A3 ... An * π / 2). How many different states of the system?

Answer 1

Answer on Question #57532 – Math - Trigonometry

Question

State of the system at any given time is determined by a set of values of $n$ parameters. Each parameter can take one of three values: 1, 0, -1. It is known that the parameter value is defined by formula $A1 = \sin(A2 * A3 \dots \text{An} * \pi / 2)$. How many different states of the system?

Solution

Find the possible states of the system for $n=1$:

$$A1 = \sin\left(\frac{\pi}{2}\right) = 1$$

The system has 1 possible state.

Find the possible states of the system for $n=2$:

$$A1 = \sin\left(A2 * \frac{\pi}{2}\right) = \begin{cases} -1, & \text{if } A2 = -1 \\ 0, & \text{if } A2 = 0 \quad \Leftrightarrow A1 = A2 \\ 1, & \text{if } A2 = 1 \end{cases}$$

The system has 3 possible states.

Find the possible states of the system for $n=3$:

$$A1 = \sin\left(A2 * A3 * \frac{\pi}{2}\right) = \begin{cases} -1, & \text{if } A2 * A3 = -1 \\ 0, & \text{if } A2 * A3 = 0 \quad \Leftrightarrow A1 = A2 * A3 \\ 1, & \text{if } A2 * A3 = 1 \end{cases}$$

The system has 9 possible states.

Find the possible states of the system for any $n$:

$$A1 = A2 * A3 * \dots * A_n$$

The system has $3^{n-1}$ possible states.

Answer: $3^{n-1}$ different states of the system.

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