Question

2 cos пx/3=1,если |x|<8

Answer 1

Cos пx\3 = 1\2
пx\3= п\3 + 2пn
пx = п + 6пn
x= 1 + 6n

Answer 2

To solve the equation 2cos(x/3) = 1, we can start by isolating cos(x/3) by dividing both sides by 2:

cos(x/3) = 1/2

Next, we need to find the values of x that satisfy this equation, given that |x| < 8. We can use the fact that cos(x/3) = 1/2 for specific angles in the unit circle, such as π/3 and 5π/3 in the first and fourth quadrants, respectively.

To find the solutions in the given interval, we need to consider all angles whose reference angles fall within the interval (-8,8). The reference angle is the acute angle between the terminal side of an angle and the x-axis.

For the first quadrant, the reference angle is x/3, so we need to solve:

x/3 = π/3 + 2πn, where n is an integer

x = π + 6πn, where n is an integer

For the fourth quadrant, the reference angle is 2π - x/3, so we need to solve:

2π - x/3 = π/3 + 2πn, where n is an integer

x = 16π/3 + 18πn, where n is an integer

However, we need to ensure that the solutions satisfy the condition |x| < 8. Since the minimum value of the solutions we obtained is π, we can check how many times we need to add 2π to π to get a value that is less than 8:

π + 2π = 3π, which is less than 8

π + 4π = 5π, which is less than 8

π + 6π = 7π, which is greater than 8

Therefore, the solutions that satisfy the given equation and the condition |x| < 8 are:

x = π, 3π, 5π, 16π/3, 22π/3, 28π/3

Note that these solutions can be simplified using common denominators.