the following involves a trigonmetric equation in quadratic form. solve the equation on the interval 0,2π)…

the following involves a trigonmetric equation in quadratic form. solve the equation on the interval 0,2π). 4tan²x - 12 = 0 select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. x = (type an exact answer in terms of π. use a comma to separate answers as needed. use integers or fractions for any numbers in the expression.) b. there is no solution.

the following involves a trigonmetric equation in quadratic form. solve the equation on the interval 0,2π). 4tan²x - 12 = 0 select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. x = (type an exact answer in terms of π. use a comma to separate answers as needed. use integers or fractions for any numbers in the expression.) b. there is no solution.

Answer

Explanation:

Step1: Solve for (\tan^{2}x)

Given (4\tan^{2}x - 12=0), add (12) to both sides: (4\tan^{2}x=12). Divide both sides by (4): (\tan^{2}x = 3).

Step2: Solve for (\tan x)

Take the square root of both sides: (\tan x=\pm\sqrt{3}).

Step3: Find (x) values in ([0, 2\pi))

When (\tan x=\sqrt{3}), (x = \frac{\pi}{3}+k\pi), (k\in\mathbb{Z}). In the interval ([0,2\pi)), (x=\frac{\pi}{3},\frac{4\pi}{3}). When (\tan x=-\sqrt{3}), (x=\frac{2\pi}{3}+k\pi), (k\in\mathbb{Z}). In the interval ([0,2\pi)), (x=\frac{2\pi}{3},\frac{5\pi}{3}).

Answer:

(x = \frac{\pi}{3},\frac{2\pi}{3},\frac{4\pi}{3},\frac{5\pi}{3})