Question

Решите уравнение 2sin^2 x+Корень из 2sinx -10sin x -5корень из двух=0 .б Помогите пожалуйста

Answer 1

Sinx=a
2a²+(√2-10)a-5√2=0
D=(√2-10)²+8*5√2=2-20√2+100+40√2=(√2+10)²
√D=√2+10
a1=(10-√2-√2-10)/4=-√2/2⇒sinx=-√2/2⇒x=(-1)^(n+1)*π/4+πn,n∈z
a2=(10-√2+√2+10)/4=5⇒sinx=5>1 нет решения

Answer 2

Solving the Equation 2sin^2(x) + √2sin(x) - 10sin(x) - 5√2 = 0

To solve the equation 2sin^2(x) + √2sin(x) - 10sin(x) - 5√2 = 0, we can follow these steps:

1. Substitute sin(x) with a variable, say u: Let u = sin(x). Then the equation becomes a quadratic equation in terms of u.

2. Solve the quadratic equation: After substituting sin(x) with u, the equation becomes 2u^2 + √2u - 10u - 5√2 = 0. We can solve this quadratic equation for u.

3. Substitute back sin(x) for u: Once we find the values of u, we can substitute them back to sin(x) to find the solutions for x.

Solution Steps:

1. Substitute sin(x) with u: Let u = sin(x). The equation becomes 2u^2 + √2u - 10u - 5√2 = 0.

2. Solve the quadratic equation: Using the quadratic formula, we can find the values of u: ``` u = (-b ± √(b^2 - 4ac)) / (2a) ```

where a = 2, b = -10 + √2, and c = -5√2.

After solving for u, we get the values of u.

3. Substitute back sin(x) for u: Once we have the values of u, we can substitute them back into sin(x) to find the solutions for x.

Conclusion:

By following these steps, we can find the solutions for the equation 2sin^2(x) + √2sin(x) - 10sin(x) - 5√2 = 0.