solve the following for x, where 0<x<2pi.... 2tanx+ sq rt 12=0

- write the general solution aswell

2 tanx = -√12

tanx = -√12/2
x = 60° or 240°
since the period of tanx is 180°
the general solution would be 60° + 180k

now I will switch to radians
x = π/3 or 4π/3
general solution : x = π/3 + kπ , where k is an integer

(I used my calculator to find the first angle, even though I should have recognized √12/2 as √3
I usually set my calculator to degrees, that way I recognize the standard angles if there are any.
Unless you recognize the decimal versions of radian angles, this works best for me)

To solve the equation 2tan(x) + √12 = 0, you need to isolate the variable x. Let's go step by step to solve it:

1. Subtract √12 from both sides of the equation to isolate the term with tan(x):
2tan(x) = -√12

2. Divide through by 2 to get:
tan(x) = -√12/2

3. Simplify the right side:
tan(x) = -√3

Now, we need to find the values of x that satisfy this equation. To do that, we can use the inverse tangent function (denoted as arctan or tan^(-1)) to find the angles whose tangent is -√3.

4. Take the inverse tangent of both sides to find the angle x:
x = arctan(-√3)

The inverse tangent will give us a principal value for x, but since we are looking for a general solution between 0 and 2π, there will be multiple solutions.

5. To find the general solution, we add or subtract any integer multiple of π to the principal value:
x = arctan(-√3) + nπ, where n is an integer.

Therefore, the general solution to the equation 2tan(x) + √12 = 0, where 0 < x < 2π, is:
x = arctan(-√3) + nπ, where n is an integer.

Note: It's important to remember that arctan only gives solutions within the range of -π/2 to π/2. To obtain solutions in the range of 0 to 2π, we introduce the integer multiple of π.