Exercises 39–52 involve trigonometric equations quadratic in form. Solve each equation on the interval [0, 2𝝅). 9 tan² x - 3 = 0

Verified step by step guidance

1. Start with the given equation: \(9 \tan^{2} x - 3 = 0\).
2. Isolate the \(\tan^{2} x\) term by adding 3 to both sides: \(9 \tan^{2} x = 3\).
3. Divide both sides by 9 to solve for \(\tan^{2} x\): \(\tan^{2} x = \frac{3}{9} = \frac{1}{3}\).
4. Take the square root of both sides to solve for \(\tan x\): \(\tan x = \pm \sqrt{\frac{1}{3}} = \pm \frac{1}{\sqrt{3}}\).
5. Find all values of \(x\) in the interval \([0, 2\pi)\) where \(\tan x = \frac{1}{\sqrt{3}}\) and \(\tan x = -\frac{1}{\sqrt{3}}\), remembering that tangent has a period of \(\pi\).

Key Concepts

Quadratic Form in Trigonometric Equations

A trigonometric equation quadratic in form involves a trigonometric function raised to the second power, such as tan²x. These can often be solved by substituting a variable (e.g., t = tan x) to transform the equation into a standard quadratic, which can then be solved using algebraic methods.

Solving Quadratic Equations

Solving quadratic equations involves finding values of the variable that satisfy the equation ax² + bx + c = 0. Methods include factoring, completing the square, or using the quadratic formula. Once the quadratic in terms of the trigonometric function is solved, the solutions are back-substituted to find the angle values.

Finding Solutions on the Interval [0, 2π)

After solving for the trigonometric function, solutions for the angle x must be found within the interval [0, 2π). This requires using the periodicity and properties of the tangent function, considering all angles in the specified range that satisfy the equation.