Question

5sin2 x/4 - 2 sin x/4 - 3 = 0 ​

Answer 1

Ответ:

Честно (это) пипец маты нельзя писать

Answer 2

To solve the equation 5sin^2(x/4) - 2sin(x/4) - 3 = 0, you can use a substitution to make it more manageable. Let's use a substitution to simplify the equation. Let y = sin(x/4).

Now, the equation becomes:

5y^2 - 2y - 3 = 0

This is a quadratic equation in terms of y. To solve for y, you can use the quadratic formula:

y = (-b ± √(b² - 4ac)) / (2a)

In this case, a = 5, b = -2, and c = -3. Plug these values into the quadratic formula:

y = (2 ± √((-2)² - 4(5)(-3))) / (2(5))

y = (2 ± √(4 + 60)) / 10

y = (2 ± √64) / 10

y = (2 ± 8) / 10

Now, you have two possible values for y:

1. y = (2 + 8) / 10 = 10/10 = 1
2. y = (2 - 8) / 10 = -6/10 = -3/5

Now that you have the values of y, recall that y = sin(x/4). So, you need to find the values of x/4 that correspond to these values of y:

1. If y = 1, then sin(x/4) = 1. This occurs when x/4 = π/2 plus any multiple of 2π. So, x/4 = π/2 + 2nπ, where n is an integer.

2. If y = -3/5, then sin(x/4) = -3/5. To find the angles corresponding to this value of sin(x/4), you can use the inverse sine function. So, x/4 = arcsin(-3/5) plus any multiple of 2π. This will give you the values of x/4.

Finally, to find the solutions for x, multiply both sides of x/4 = π/2 + 2nπ or x/4 = arcsin(-3/5) + 2nπ by 4 to get the solutions for x:

1. x = 2π + 8nπ, where n is an integer.
2. x = 4arcsin(-3/5) + 8nπ, where n is an integer.

These are the solutions to the equation 5sin^2(x/4) - 2sin(x/4) - 3 = 0.