Cos(4x+п/4)=-корень 2/2 одз: [-П;п)

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вопрос

To solve the equation cos(4x + π/4) = -√2/2 on the interval [-π, π), we can use the inverse cosine function.

First, we need to isolate the cosine function by taking the inverse cosine of both sides:

4x + π/4 = arccos(-√2/2)

Now, we need to find the angle whose cosine is -√2/2. It is important to note that the cosine function is negative in the second and third quadrants.

The angle in the second quadrant with cosine equal to -√2/2 is 3π/4, while the angle in the third quadrant is 5π/4.

Therefore, we have two possible solutions for arccos(-√2/2): 3π/4 and 5π/4.

Solving for x:

For arccos(-√2/2) = 3π/4:

4x + π/4 = 3π/4
4x = 2π/4
x = π/2

For arccos(-√2/2) = 5π/4:

4x + π/4 = 5π/4
4x = 4π/4
x = π

Therefore, the solutions on the interval [-π, π) are x = π/2 and x = π.