Exercises 39–52 involve trigonometric equations quadratic in form. Solve each equation on the interval [0, 2𝝅). 4 cos² x - 1 = 0

Verified step by step guidance

1. Recognize that the equation is quadratic in form with respect to \(\cos x\). Rewrite the equation as \(4 \cos^{2} x - 1 = 0\).
2. Isolate the squared term by adding 1 to both sides: \(4 \cos^{2} x = 1\).
3. Divide both sides by 4 to solve for \(\cos^{2} x\): \(\cos^{2} x = \frac{1}{4}\).
4. Take the square root of both sides to solve for \(\cos x\): \(\cos x = \pm \frac{1}{2}\).
5. Find all values of \(x\) in the interval \([0, 2\pi)\) where \(\cos x = \frac{1}{2}\) and where \(\cos x = -\frac{1}{2}\). Use the unit circle or inverse cosine function to determine these angles.

Key Concepts

Quadratic Form in Trigonometric Equations

A trigonometric equation is quadratic in form when it can be expressed similarly to a quadratic equation, such as involving terms like cos²x or sin²x. Recognizing this allows you to use substitution methods or algebraic techniques to solve for the trigonometric function before finding the angle solutions.

Solving Basic Trigonometric Equations

Once the quadratic form is simplified, solving for the trigonometric function (e.g., cos x) involves isolating the function and finding all angles within the given interval that satisfy the equation. This requires knowledge of the unit circle and the values of trigonometric functions at standard angles.

Interval Restriction and Solution Sets

The problem restricts solutions to the interval [0, 2π), meaning only angles within one full rotation are considered. After finding possible values for the trigonometric function, you must determine all corresponding angles in this interval, accounting for the periodic nature of trigonometric functions.