In Exercises 85–96, use a calculator to solve each equation, correct to four decimal places, on the interval [0, 2𝝅). 4 tan² x - 8 tan x + 3 = 0

Verified step by step guidance

1. Recognize that the given equation is a quadratic in terms of \( \tan x \): \(4 \tan^{2} x - 8 \tan x + 3 = 0\). Our goal is to solve for \(x\) in the interval \([0, 2\pi)\).
2. Let \(t = \tan x\). Rewrite the equation as \(4t^{2} - 8t + 3 = 0\). This is a standard quadratic equation in \(t\).
3. Use the quadratic formula to solve for \(t\): \[ t = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \] where \(a=4\), \(b=-8\), and \(c=3\).
4. Calculate the two values of \(t\) from the quadratic formula. Each value corresponds to \(\tan x = t\).
5. For each value of \(t\), find the solutions for \(x\) by taking the arctangent: \[ x = \arctan(t) \]. Since \(\tan x\) is periodic with period \(\pi\), add \(\pi\) to each principal solution to find all solutions in \([0, 2\pi)\). Use a calculator to find the values of \(x\) correct to four decimal places.

Key Concepts

Solving Quadratic Equations in Trigonometric Functions

This involves treating the trigonometric expression, such as tan x, as a variable and solving the resulting quadratic equation using algebraic methods like factoring or the quadratic formula. The solutions for tan x are then used to find the corresponding angles x.

Using the Tangent Function and Its Properties

The tangent function relates an angle in a right triangle to the ratio of the opposite side over the adjacent side. It is periodic with period π, meaning solutions repeat every π radians. Understanding its domain and range helps identify valid solutions within the given interval.

Finding Solutions on a Specified Interval with a Calculator

After solving for tan x, use a calculator to find the inverse tangent values and adjust for all solutions within the interval [0, 2π). Since tan x has period π, two solutions may exist per period. Round answers to the required decimal places as specified.