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 equation is a quadratic in terms of \( \tan x \). It can be rewritten as \( 4u^2 - 8u + 3 = 0 \) where \( u = \tan x \).
2. Use the quadratic formula \( u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to solve for \( u \), where \( a = 4 \), \( b = -8 \), and \( c = 3 \).
3. Calculate the discriminant \( b^2 - 4ac \) to determine the nature of the roots.
4. Substitute the values of \( a \), \( b \), and \( c \) into the quadratic formula to find the possible values of \( u = \tan x \).
5. For each solution \( u \), use the inverse tangent function \( x = \tan^{-1}(u) \) to find the angle \( x \) in the interval \([0, 2\pi)\).

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate angles to ratios of sides in right triangles. The tangent function, specifically, is defined as the ratio of the opposite side to the adjacent side. Understanding how to manipulate and solve equations involving these functions is crucial for solving trigonometric equations.

Quadratic Equations

The equation given, 4 tan² x - 8 tan x + 3 = 0, is a quadratic equation in terms of tan x. Quadratic equations can be solved using various methods, including factoring, completing the square, or applying the quadratic formula. Recognizing the structure of the equation allows for the appropriate method to find the solutions for tan x.

Interval Notation

The interval [0, 2𝝅) specifies the range of values for x in which we are interested in finding solutions. This notation indicates that x can take any value from 0 up to, but not including, 2𝝅. Understanding interval notation is essential for determining the valid solutions of trigonometric equations, as trigonometric functions are periodic and can have multiple solutions.