Question

Arcsin 3x/5 + arcsin 4x/5 = arcsin x

Answer 1

Ответ:

x = -1; 0; 1

Объяснение:

arcsin 3x/5 + arcsin 4x/5 = arcsin x

sin(arcsin 3x/5 + arcsin 4x/5) = sin arcsin x

3x/5 · cos arcsin 4x/5 + cos arcsin 3x/5 · 4x/5 = x

x · (3/5 · cos arcsin 4x/5 + 4/5 · cos arcsin 3x/5 - 1) = 0

x = 0

3/5 · cos arcsin 4x/5 + 4/5 · cos arcsin 3x/5 - 1 = 0

3/5 · cos arcsin 4x/5 + 4/5 · cos arcsin 3x/5 - 1 = 0

3/5 · √(1 - sin²arcsin 4x/5) + 4/5 · √(1 - sin²arcsin 3x/5) = 1

3 · √(1 - (4x/5)²) + 4 · √(1 - (3x/5)²) = 5

3 · √(25 - 16x²) + 4 · √(25 - 9x²) = 25

9 · (25 - 16x²) + 24 · √((25 - 16x²)(25 - 9x²)) + 16 · (25 - 9x²) = 625

225 - 144x² + 24 · √(625 - 225x² - 400x² + 144x⁴) + 400 - 144x² = 625

625 - 288x² + 24 · √(625 - 625x² + 144x⁴) = 625

24 · √(625 - 625x² + 144x⁴) = 288x²

√(625 - 625x² + 144x⁴) = 12x²

625 - 625x² + 144x⁴ = 144x⁴

625 · (1 - x) · (1 + x) = 0

1 - x = 0

1 + x = 0

x = 1

x = -1

Answer 2

To solve the equation arcsin(3x/5) + arcsin(4x/5) = arcsin(x), we can make use of the trigonometric identity for the sum of two arcsines:

arcsin(a) + arcsin(b) = arcsin(a√(1 - b²) + b√(1 - a²)).

Using this identity, we can rewrite the equation as follows:

arcsin(3x/5) + arcsin(4x/5) = arcsin(x)

=> arcsin(3x/5) + arcsin(4x/5) = arcsin(x√(1 - (3x/5)²) + (4x/5)√(1 - x²)).

Now, let's simplify the equation step by step:

arcsin(3x/5) + arcsin(4x/5) = arcsin(x√(1 - (3x/5)²) + (4x/5)√(1 - x²))

=> arcsin(3x/5) + arcsin(4x/5) = arcsin(x√(1 - 9x²/25) + (4x/5)√(1 - x²))

=> arcsin(3x/5) + arcsin(4x/5) = arcsin(x√(25 - 9x²)/5 + (4x/5)√(25 - 25x²)/5)

=> arcsin(3x/5) + arcsin(4x/5) = arcsin((5x√(25 - 9x²) + 4x√(25 - 25x²))/5)

Now, comparing both sides of the equation, we can deduce that:

(3x/5) = (5x√(25 - 9x²) + 4x√(25 - 25x²))/5

Multiplying both sides of the equation by 5:

3x = 5x√(25 - 9x²) + 4x√(25 - 25x²)

Now, let's isolate the radical terms:

3x - 5x√(25 - 9x²) = 4x√(25 - 25x²)

Squaring both sides of the equation to eliminate the radicals:

(3x - 5x√(25 - 9x²))² = (4x√(25 - 25x²))²

Expanding and simplifying:

9x² - 30x²√(25 - 9x²) + 25x²(25 - 9x²) = 16x²(25 - 25x²)

Combining like terms:

9x² + 25x²(25 - 9x²) - 30x²√(25 - 9x²) = 16x²(25 - 25x²)

Expanding further:

9x² + 625x² - 225x⁴ - 30x²√(25 - 9x²) = 400x² - 400x⁴

Rearranging the terms:

225x⁴ - 40x² + 30x²√(25 - 9x²) - 625x² = 0

225x⁴ - 635x² + 30x²√(25 - 9x²) = 0

Now,