Question

2cosx(сosx -cos 5п/4)+cosx+cos 3п/4=0

Answer 1

Открываем скобки: 2cosx(cosx+(sqr2)/2)+cosx-(sqr2)/2)=0  2cos^2(x)+sqr(2)cosx+cosx-sqr(2)/2=0   2cos^2(x)+(sqr(2)+1)cosx-sqr(2)/2=0   внесём изменения cosx=a у нас получается квадратные уравнения 2а^2+(sqr(2)+1)a-sqr(2)/2=0

Answer 2

To solve the equation 2cos(x)(cos(x) - cos(5π/4)) + cos(x) + cos(3π/4) = 0, we'll simplify and solve step by step.

Let's simplify the equation:

2cos(x)(cos(x) - cos(5π/4)) + cos(x) + cos(3π/4) = 0

Expand the terms: 2cos(x)cos(x) - 2cos(x)cos(5π/4) + cos(x) + cos(3π/4) = 0

Simplify the cosine terms: 2cos²(x) - 2cos(x)cos(5π/4) + cos(x) + cos(3π/4) = 0

Recall the trigonometric identity: cos(θ - π/4) = cos(θ)cos(π/4) + sin(θ)sin(π/4) We'll use this identity to simplify the equation further.

Substitute cos(5π/4) and cos(3π/4) using the identity:

2cos²(x) - 2cos(x)[cos(x)cos(π/4) + sin(x)sin(π/4)] + cos(x) + [cos(x)cos(π/4) - sin(x)sin(π/4)] = 0

Simplify the equation using the trigonometric identity:

2cos²(x) - 2cos²(x)cos(π/4) - 2sin(x)sin(π/4)cos(x) + cos(x) + cos(x)cos(π/4) - sin(x)sin(π/4) = 0

Now let's simplify further:

2cos²(x) - 2cos²(x)cos(π/4) + cos(x)cos(π/4) + cos(x) - 2sin(x)sin(π/4)cos(x) - sin(x)sin(π/4) = 0

Combine like terms:

2cos²(x) - 2cos²(x)cos(π/4) + cos(x)cos(π/4) + cos(x) - sin(x)[2sin(π/4)cos(x) + sin(π/4)] = 0

Apply the trigonometric identity sin(π/4) = cos(π/4) = sqrt(2)/2:

2cos²(x) - 2cos²(x)(sqrt(2)/2) + cos(x)(sqrt(2)/2) + cos(x) - sin(x)[2(sqrt(2)/2)cos(x) + sqrt(2)/2] = 0

Simplify the terms:

2cos²(x) - sqrt(2)cos²(x) + (sqrt(2)/2)cos(x) + cos(x) - sin(x)[sqrt(2)cos(x) + sqrt(2)/2] = 0

Combine like terms:

(2 - sqrt(2))cos²(x) + (sqrt(2)/2 + 1)cos(x) - sin(x)(sqrt(2)cos(x) + sqrt(2)/2) = 0

Now, let's focus on solving for x.

The equation is now in terms of trigonometric functions and is not easily solved analytically. To find the numerical solutions, you can use numerical methods or graphing techniques.

One approach is to use numerical methods such as the Newton-Raphson method or the bisection method to approximate the solutions. Alternatively, you can plot the equation on a graphing calculator or software to visualize the solutions.

Keep in mind that this equation may have multiple solutions depending on the interval or range of x values you are considering.