Find the general solution for f(x) = 27 tan^2 x-9=0

27tan^2 x = 9

tan^2 x = 1/3
tanx = ± 1/√3
x = 30° , 150° , 210° or 330°

general
x = (30 + 180k)° , where k is an integer
x = (150+180k)° , where k is an integer

To find the general solution for the equation f(x) = 27 tan^2 x - 9 = 0, we need to solve the equation for x. Let's break the problem down step by step:

Step 1: Simplify the equation
Start by simplifying the equation. In this case, we can simplify 27 tan^2 x - 9 = 0 as follows:

27 tan^2 x - 9 = 0
Divide the entire equation by 9 to simplify:
3 tan^2 x - 1 = 0

Step 2: Substitute tan^2 x with sin^2 x / cos^2 x
Using the identity tan^2 x = sin^2 x / cos^2 x, we can substitute tan^2 x in the equation as follows:

3 (sin^2 x / cos^2 x) - 1 = 0

Step 3: Multiply through by cos^2 x
To eliminate the denominator, multiply both sides of the equation by cos^2 x:

3 sin^2 x - cos^2 x = 0

Step 4: Use the Pythagorean Identity
Using the Pythagorean Identity sin^2 x + cos^2 x = 1, we can rewrite the equation as:

3 sin^2 x - (1 - sin^2 x) = 0
3 sin^2 x - 1 + sin^2 x = 0
4 sin^2 x - 1 = 0

Step 5: Solve for sin^2 x
To solve for sin^2 x, isolate the term by moving -1 to the other side:

4 sin^2 x = 1
Divide both sides of the equation by 4:

sin^2 x = 1/4

Step 6: Take the square root of both sides
To solve for sin x, take the square root of both sides of the equation:

sin x = ±√(1/4)
sin x = ±1/2

Step 7: Find the angles
To find the angles that satisfy the equation, we can look at the unit circle. The sine function is positive in the first and second quadrants, where the y-coordinate is positive (1/2 in this case).

So, the two possible values for x are:

x = π/6 + 2πn, where n is an integer (for the first quadrant)
x = 5π/6 + 2πn, where n is an integer (for the second quadrant)

Therefore, the general solution for f(x) = 27 tan^2 x - 9 = 0 is:

x = π/6 + 2πn, 5π/6 + 2πn, where n is an integer.