Question

Sin4x + sin^2 2x=0 решите пожалуйста, буду очень благодарна

Answer 1

Sin(4x) + sin^2(2x) = 0
sin(2x)*(2cos(2x) - sin(2x)) = 0
1)  sin(2x) = 0
2x = πn, n∈Z
x1 = (πn)/2, n∈Z
2) 2cos(2x) - sin(2x) = 0             / cos(2x) ≠ 0
2 - tg(2x) = 0
tg(2x) = 2
2x = arctg2 + πk, k∈Z
x2 = (1/2)*arctg2 + (1/2)πk, k∈Z

Answer 2

Solving the Equation sin(4x) + sin^2(2x) = 0

To solve the equation sin(4x) + sin^2(2x) = 0, we can use trigonometric identities and algebraic manipulation to simplify and solve for the values of x that satisfy the equation.

Using Trigonometric Identities

We can start by using the trigonometric identity sin^2(x) = (1 - cos(2x))/2 to rewrite sin^2(2x) in terms of cos(2x). This will help us simplify the equation.

Rewriting the Equation

The equation sin(4x) + sin^2(2x) = 0 can be rewritten as sin(4x) + (1 - cos(4x))/2 = 0, using the identity sin^2(x) = (1 - cos(2x))/2.

Simplifying the Equation

Next, we can simplify the equation further by multiplying through by 2 to clear the fraction, resulting in 2sin(4x) + (1 - cos(4x)) = 0.

Using Double-Angle Identity

We can then use the double-angle identity sin(2x) = 2sin(x)cos(x) to express sin(4x) in terms of sin(2x) and cos(2x).

Substituting Double-Angle Identity

Substituting sin(4x) = 2sin(2x)cos(2x) into the equation gives 2(2sin(2x)cos(2x)) + (1 - cos(4x)) = 0.

Solving for x

By substituting sin(4x) = 2sin(2x)cos(2x) and simplifying the equation, we can solve for the values of x that satisfy the equation.

Please note that the solution process involves multiple steps of trigonometric identity manipulation and algebraic simplification.

Let's proceed with the solution.