Question

Cos x = - 1/2........​

Answer 1

$x = + - arccos( - \frac{1}{2} ) + 2\pi \: k \\ x = + - (\pi - arccos \frac{1}{2} ) + 2\pi \: k \\ x = + - \frac{2\pi}{3} + 2\pi \: k$

к принадлежит множеству целых чисел

Answer 2

To find the solutions for the equation cos(x) = -1/2, we need to determine the angles whose cosine value is -1/2. In the unit circle, the cosine function represents the x-coordinate of a point on the circle.

The unit circle is a circle with a radius of 1, centered at the origin (0, 0) in the coordinate plane.

For cos(x) = -1/2, there are two angles in the interval [0, 2π] that satisfy this equation. These angles are in the second and third quadrants where the x-coordinate is negative:

1. First solution (in the second quadrant): To find the first solution, we use the inverse cosine function (arccos or cos^(-1)) and evaluate it for -1/2:

x₁ = arccos(-1/2) ≈ 2.094 radians ≈ 120 degrees (rounded to the nearest degree).

2. Second solution (in the third quadrant): For the second solution, we can use the symmetry of the unit circle in the third quadrant:

x₂ = π - x₁ ≈ 3.142 - 2.094 ≈ 1.048 radians ≈ 60 degrees (rounded to the nearest degree).

So, the solutions for the equation cos(x) = -1/2 are approximately x = 2.094 radians (or 120 degrees) and x = 1.048 radians (or 60 degrees). Keep in mind that there are infinitely many solutions to this equation, but these are the primary solutions within the interval [0, 2π].