Question

Sin²(2x)=cos2x+4sin⁴x

Answer 1

Тебе осталось найти объединение, если вас об этом просят

Answer 2

To solve the equation sin²(2x) = cos(2x) + 4sin⁴(x), we'll first use trigonometric identities to simplify the equation and then find the solutions for x.

Step 1: Utilize trigonometric identities

1. sin²(2x) = 2sin(x)cos(x)
2. cos(2x) = cos²(x) - sin²(x) = 1 - 2sin²(x) (using double-angle formula)
3. 4sin⁴(x) = 4(sin²(x))²

Step 2: Rewrite the equation using the simplified expressions Now, the equation becomes: 2sin(x)cos(x) = 1 - 2sin²(x) + 4(sin²(x))²

Step 3: Simplify the equation further To get rid of the cosine term, we can use the identity: 2sin(x)cos(x) = sin(2x) So, the equation becomes: sin(2x) = 1 - 2sin²(x) + 4(sin²(x))²

Step 4: Let y = sin(x) The equation can be rewritten in terms of y: sin(2x) = 1 - 2y² + 4y⁴

Step 5: Solve the quadratic equation in terms of y 2y² - 4y⁴ = 1 - sin(2x)

Step 6: Set the equation to zero to find the roots 2y² - 4y⁴ - (1 - sin(2x)) = 0

Step 7: Solve the quartic equation for y This is a quartic equation in terms of y. Solving quartic equations can be quite complex, and in this case, the algebraic solution might not be straightforward. It can be time-consuming and cumbersome to solve manually.

Therefore, if you are looking for numerical solutions for specific values of x, you can use numerical methods or graphing techniques to find approximate solutions.

Keep in mind that there might be multiple solutions for y (sin(x)), which correspond to different values of x, due to the periodic nature of trigonometric functions. Additionally, some solutions may involve complex numbers, depending on the values of sin(2x) in the range [-1, 1].