Question

√3 ctg x = 2 | cos x |

Answer 1

Ответ:

1 ctgx=-√3/3 x=2π/3+πn,n∈z 2 sinx-2cosx=0/cosx tgx-2=0 tgx=2 x=arctg2+πn,n∈z

Пошаговое объяснение:

1 ctgx=-√3/3 x=2π/3+πn,n∈z 2 sinx-2cosx=0/cosx tgx-2=0 tgx=2 x=arctg2+πn,n∈z

Answer 2

To solve the equation √3 ctg x = 2 |cos x|, we'll break it down step by step.

First, let's simplify the equation using trigonometric identities:

ctg x is the same as cot x, which is equal to 1/tan x. So, we can rewrite the equation as:

√3 / tan x = 2 |cos x|

Next, we'll rewrite |cos x| using its definition:

|cos x| = cos x if cos x ≥ 0 |cos x| = -cos x if cos x < 0

Now, we'll consider the two cases separately:

Case 1: cos x ≥ 0

In this case, |cos x| = cos x, so our equation becomes:

√3 / tan x = 2 cos x

We can rewrite tan x as sin x / cos x:

√3 / (sin x / cos x) = 2 cos x

Multiply both sides by cos x to eliminate the denominator:

√3 = 2 sin x

Divide both sides by 2:

√3 / 2 = sin x

Taking the arcsin of both sides:

x = arcsin(√3 / 2)

Since arcsin(√3 / 2) = π/3, we have:

x = π/3 + 2kπ, where k is an integer.

Case 2: cos x < 0

In this case, |cos x| = -cos x, so our equation becomes:

√3 / tan x = 2 (-cos x)

√3 / tan x = -2 cos x

Following the same steps as in Case 1, we arrive at:

√3 / 2 = sin x

Taking the arcsin of both sides:

x = arcsin(√3 / 2)

Since arcsin(√3 / 2) = π/3, we have:

x = π - π/3 + 2kπ = 2π/3 + 2kπ, where k is an integer.

So, the general solutions to the equation are:

x = π/3 + 2kπ, where k is an integer, and x = 2π/3 + 2kπ, where k is an integer.