Question

2tg(2π-x)+ctg(x-3π/2)=√3

Answer 1

По формулам приведения :

$2{\rm tg}(2\pi -x)+{\rm ctg}(x-\frac{3\pi}{2})=\sqrt{3}\\ \\ -2{\rm tg}\, x+{\rm tg}\, x=\sqrt{3}\\ \\ {\rm tg}\, x=-\sqrt{3}\\ \\ x=-\frac{\pi}{3}+\pi n,n \in \mathbb{Z}$

Answer 2

To solve the trigonometric equation 2tg(2π-x) + ctg(x-3π/2) = √3, we'll first rewrite ctg(x) in terms of cotangent. The cotangent function (ctg) is the reciprocal of the tangent function (tan), so ctg(x) = 1/tan(x). Therefore, we can rewrite the equation as follows:

2tan(2π-x) + cot(3π/2 - x) = √3.

Next, we'll simplify the trigonometric expressions using the periodicity and symmetry of the tangent and cotangent functions. Remember that tan(π - x) = -tan(x) and cot(π - x) = -cot(x):

2tan(-x) - cot(x - π/2) = √3.

Now, we'll use the trigonometric identity tan(-x) = -tan(x):

-2tan(x) - cot(x - π/2) = √3.

Now, let's express cot(x - π/2) in terms of tangent:

cot(x - π/2) = 1/tan(x - π/2).

Since tan(x - π/2) is the tangent of the angle whose reference angle is (x - π/2), we can use the identity tan(θ) = 1/tan(π/2 - θ):

cot(x - π/2) = tan(π/2 - (x - π/2)) = tan(π/2 - x).

So the equation becomes:

-2tan(x) - tan(π/2 - x) = √3.

Next, we'll use the tangent addition formula: tan(A - B) = (tan(A) - tan(B)) / (1 + tan(A) * tan(B)):

tan(π/2 - x) = (tan(π/2) - tan(x)) / (1 + tan(π/2) * tan(x)).

Remember that tan(π/2) is undefined, so tan(π/2) is infinity. The equation now becomes:

-2tan(x) - [(infinity - tan(x)) / (1 + infinity * tan(x))] = √3.

We can simplify the equation further by considering that any value multiplied by infinity is still infinity, and any finite value divided by infinity is approaching zero:

-2tan(x) + tan(x) = √3.

Now combine the terms on the left side:

-tan(x) = √3.

Finally, solve for tan(x):

tan(x) = -√3.

Now, let's find the solutions for x. Remember that tangent is negative in the third and fourth quadrants.

Using the unit circle, in the third quadrant, we have:

x = π + arctan(-√3).

And in the fourth quadrant, we have:

x = 2π + arctan(-√3).

Now, calculate the values:

x = π + (-π/3) = 2π/3,

and

x = 2π + (-π/3) = 5π/3.

So the solutions to the equation are x = 2π/3 and x = 5π/3.