Question

Решите уравнение: arcsin x · arccos x = −1;

Answer 1

$\sin(2x ) \div 2 = - 1 \\ \sin(2x) = - 2$

Answer 2

To solve the equation arcsin(x) · arccos(x) = -1, we need to find the value of x that satisfies the equation. Let's proceed with the solution step by step.

Step 1: Recognize the domain of the functions arcsin(x) and arccos(x). The domain of arcsin(x) is [-1, 1], and the domain of arccos(x) is also [-1, 1].

Step 2: Write the equation. arcsin(x) · arccos(x) = -1

Step 3: Use the definition of inverse trigonometric functions. The arcsin(x) function returns an angle whose sine is x, and the arccos(x) function returns an angle whose cosine is x.

Step 4: Find the value of x. Since arcsin and arccos are inverses of each other, we can rewrite the equation as follows: arcsin(arccos(x)) = -1

Step 5: Solve for arccos(x) first. Since the range of arccos(x) is [0, π], the only solution for the equation arcsin(arccos(x)) = -1 is when arccos(x) = cos(-1) = π.

Step 6: Solve for x. Now that we know arccos(x) = π, we can find x using the definition of arccos: x = cos(π)

However, there is a problem here. The value of cos(π) is -1, but remember that the domain of arccos(x) is [0, π]. Therefore, x = -1 is not a valid solution.

Step 7: No solution. Since there is no valid solution for x in the domain of both arcsin(x) and arccos(x), the given equation arcsin(x) · arccos(x) = -1 has no solution.

Final Answer: The equation has no solution in the given domain.