Solve each equation over the interval [0, 2π). Write solutions as exact values or to four decimal places, as appropriate
tan 2x + sec 2x = 3

Verified step by step guidance

1. Start with the given equation: \(\tan 2x + \sec 2x = 3\).
2. Recall the definitions: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) and \(\sec \theta = \frac{1}{\cos \theta}\). Substitute these into the equation to get \(\frac{\sin 2x}{\cos 2x} + \frac{1}{\cos 2x} = 3\).
3. Combine the terms over the common denominator \(\cos 2x\): \(\frac{\sin 2x + 1}{\cos 2x} = 3\).
4. Multiply both sides by \(\cos 2x\) to clear the denominator: \(\sin 2x + 1 = 3 \cos 2x\).
5. Rewrite the equation as \(\sin 2x - 3 \cos 2x = -1\) and use the identity for a linear combination of sine and cosine: \(a \sin \theta + b \cos \theta = R \sin(\theta + \alpha)\), where \(R = \sqrt{a^2 + b^2}\) and \(\alpha = \arctan(\frac{b}{a})\). Apply this to express the left side as a single sine function and then solve for \(x\) over \([0, 2\pi)\).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. For this problem, knowing the relationship between tangent and secant, such as sec²θ = 1 + tan²θ, helps in transforming and simplifying the equation to a solvable form.

Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and finding all angle solutions within the given interval. This often requires algebraic manipulation, use of identities, and considering the periodicity of functions to find all valid solutions between 0 and 2π.

Interval and Exact Values

The problem restricts solutions to the interval [0, 2π), meaning all solutions must be found within one full rotation of the unit circle. Solutions should be expressed as exact values (like π/4) or decimal approximations to four decimal places, ensuring clarity and precision in the final answers.