Question

Y = √(cosx - 1/2)найдите область определения

Answer 1

cosx-1/2≥0

cosx≥1/2

есть формула для решения неравенств

cosx≥a (IaI<1) x∈[-arccosa+2πn;arccosa+2πn], n∈Z

a=1/2

x∈[-arrcos(1/2)+2πn;arccos(1/2)+2πn]

x∈[(-√3)/2+2πn;(√3)/2+2πn], n∈Z

еще можно решить графически но так проще

Answer 2

To find the domain (область определения) of the function Y = √(cos(x) - 1/2), we need to identify the values of x for which the function is defined.

The square root function (√) is defined only for non-negative real numbers. In the expression √(cos(x) - 1/2), the argument inside the square root (cos(x) - 1/2) must be greater than or equal to zero to ensure the function is defined.

So, we set the argument to be greater than or equal to zero and solve for x:

cos(x) - 1/2 ≥ 0

Now, isolate cos(x):

cos(x) ≥ 1/2

To find the values of x that satisfy this inequality, we need to consider where the cosine function is greater than or equal to 1/2. The cosine function has values between -1 and 1, with its maximum value being 1 when the angle is 0 degrees or any multiple of 2π radians.

The cosine function is symmetric about the y-axis, so its value at -θ is the same as its value at θ. Since the cosine function repeats its values every 2π radians, we need to find all the angles between 0 and 2π for which the cosine is greater than or equal to 1/2.

One such angle is π/3 (60 degrees) since cos(π/3) = 1/2. Additionally, another angle would be 5π/3 (300 degrees) since cos(5π/3) = cos(-π/3) = 1/2.

So, the domain of the function Y = √(cos(x) - 1/2) is the set of all real numbers x for which:

x ∈ [π/3, 5π/3] + 2πk, where k is an integer.

This interval represents all possible values of x for which the expression inside the square root is non-negative, making the function Y defined for these values of x.