Question

Ctg(x/2+π/8)=√ 3 help

Answer 1

Ответ:

Объяснение:

ctg(x/2+π/8)=√3

x/2+π/8=π/6+πn

x/2=π/6-π/8+πn  |×2

x=π/3-π/4+2πn=(4π-3π)/12+2πn=π/12+2πn

x=π/12+2πn.

Answer 2

To solve the trigonometric equation, we have:

Ctg(x/2 + π/8) = √3

First, let's simplify the expression on the left-hand side. The cotangent function can be expressed in terms of the tangent function:

Ctg(x/2 + π/8) = 1 / Tan(x/2 + π/8)

Next, let's use the angle sum formula for tangent:

Tan(a + b) = (Tan(a) + Tan(b)) / (1 - Tan(a) * Tan(b))

In this case, a = x/2 and b = π/8:

Tan(x/2 + π/8) = (Tan(x/2) + Tan(π/8)) / (1 - Tan(x/2) * Tan(π/8))

Now, we need to find the values of Tan(x/2) and Tan(π/8). Let's compute them separately:

1. Tan(x/2): We can use the half-angle formula for tangent: Tan(x/2) = (1 - Cos(x)) / Sin(x)

2. Tan(π/8): To find Tan(π/8), we first need to determine the values of Cos(π/8) and Sin(π/8): Cos(π/8) = √(2 + √3) / 2 Sin(π/8) = √(2 - √3) / 2

Then, we can find Tan(π/8) using the tangent definition: Tan(π/8) = Sin(π/8) / Cos(π/8)

Now, we can substitute these expressions back into the original equation:

Ctg(x/2 + π/8) = 1 / Tan(x/2 + π/8) Ctg(x/2 + π/8) = 1 / [(Tan(x/2) + Tan(π/8)) / (1 - Tan(x/2) * Tan(π/8))]

Ctg(x/2 + π/8) = (1 - Tan(x/2) * Tan(π/8)) / (Tan(x/2) + Tan(π/8))

Now, let's substitute the values we found earlier:

Tan(x/2) = (1 - Cos(x)) / Sin(x) Tan(π/8) = Sin(π/8) / Cos(π/8)

Ctg(x/2 + π/8) = (1 - [(1 - Cos(x)) / Sin(x)] * [Sin(π/8) / Cos(π/8)]) / ([(1 - Cos(x)) / Sin(x)] + [Sin(π/8) / Cos(π/8)])

Simplifying further:

Ctg(x/2 + π/8) = (Sin(x) * Cos(π/8) - (1 - Cos(x)) * Sin(π/8)) / (Sin(x) * Cos(π/8) + Sin(π/8) * Cos(x))

Now, we know that Ctg(x/2 + π/8) = √3, so we can set up the equation:

(Sin(x) * Cos(π/8) - (1 - Cos(x)) * Sin(π/8)) / (Sin(x) * Cos(π/8) + Sin(π/8) * Cos(x)) = √3

Unfortunately, we can't further simplify the equation without knowing more information about the value of x. To find the specific value(s) of x that satisfy the equation, we would need additional context or constraints on the domain of x.