Genral solution of:

2tan x + 3=0

Solve for x if:
2tan x +3 =0 and x E[-180;180]

2tanx = -3

tanx = -3/2
x must be in quadrants II or IV
angle in standard position , tan^-1 (3/2) = 56.3°

so x = 180-56.3 = 123.7° or
x = 360 - 56.3 = 303.7°

general solution
x = 123.7 + 180k or x = 303.7 + 180k, where k is an integer.

for the second part, 303.7 is clearly too large
but 123.7-180 = -56.3° will work

check: 2tan (-56.3) + 3 = 0.0011 , close enough

To find the general solution of the equation 2tan x + 3 = 0, we need to isolate the variable x.

First, let's subtract 3 from both sides of the equation:
2tan x + 3 - 3 = 0 - 3
2tan x = -3

Next, divide both sides of the equation by 2 to isolate the tangent function:
(2tan x) / 2 = (-3) / 2
tan x = -3/2

Now, to find the general solution, we need to find all possible angles that satisfy the equation. The tangent function repeats itself every π radians or 180 degrees. So, we'll find the reference angle that satisfies tan x = -3/2 and then generalize the solution based on that reference angle.

From the given equation tan x = -3/2, we can determine that x is in the third and fourth quadrants, where the tangent is negative.

To find the reference angle (θ), we can use the inverse tangent function (or arctan) which gives the angle whose tangent is a given value. Using a calculator, we can find the reference angle:

θ = arctan(-3/2) ≈ -56.31 degrees or approximately -0.98279 radians

Since θ is negative, we add π (180 degrees) or π radians to it to find the next angle in the third quadrant:
x1 = θ + π ≈ -0.98279 + π

Similarly, add 180 degrees or π radians to θ to find the angle in the fourth quadrant:
x2 = θ + π ≈ -0.98279 + 2π

So, the general solution for the equation 2tan x + 3 = 0, with x E[-180;180], is:
x = x1, x2 ≈ -0.98279 + π, -0.98279 + 2π (in radians)
x = x1, x2 ≈ -0.98279 + 180, -0.98279 + 360 (in degrees)