Question

Решить уравнение cos2x=1/8

Answer 1

$cos2x=\dfrac{1}{8} \\\\\ 2x=\pm arccos\frac{1}{8} +2\pi n \\ \boxed {x=\dfrac{\pm arccos\frac{1}{8} }{2}+ \pi n}$

Answer 2

To solve the equation cos(2x) = 1/8, we'll use trigonometric identities and algebraic manipulation. The double angle formula for cosine states:

cos(2x) = 2 * cos^2(x) - 1

So, the equation can be rewritten as:

2 * cos^2(x) - 1 = 1/8

Now, let's solve for cos^2(x):

2 * cos^2(x) = 1/8 + 1 cos^2(x) = (1 + 1/8) / 2 cos^2(x) = 9/16

Now, find the values of cos(x) by taking the square root:

cos(x) = ± sqrt(9/16)

cos(x) = ± (3/4)

There are two possible solutions for cos(x):

1. cos(x) = 3/4
2. cos(x) = -3/4

Now, we need to find the values of x. To do that, use the inverse cosine function (cos^(-1)):

1. x = cos^(-1)(3/4)
2. x = cos^(-1)(-3/4)

Now, calculate the values of x using a calculator:

1. x ≈ 0.7227 rad (rounded to four decimal places)
2. x ≈ 2.4189 rad (rounded to four decimal places)

Since cosine has a period of 2π (360 degrees), you can find other solutions by adding or subtracting multiples of 2π:

For the first solution: x = 0.7227 + 2π * n, where n is an integer (positive or negative).

For the second solution: x = 2.4189 + 2π * m, where m is an integer (positive or negative).

These are the general solutions for the given equation.