Solve the following equations.
$\(\tan\)^22\(\theta\)=1,0\(\lt{\theta}\]\lt{\pi}\)$

Verified step by step guidance

1. Recognize that the equation \( \tan^2(2\theta) = 1 \) implies \( \tan(2\theta) = \pm 1 \).
2. Consider the principal values for \( \tan(2\theta) = 1 \), which occur at \( 2\theta = \frac{\pi}{4} + n\pi \) for integer \( n \).
3. Consider the principal values for \( \tan(2\theta) = -1 \), which occur at \( 2\theta = \frac{3\pi}{4} + n\pi \) for integer \( n \).
4. Solve for \( \theta \) by dividing each equation by 2: \( \theta = \frac{\pi}{8} + \frac{n\pi}{2} \) and \( \theta = \frac{3\pi}{8} + \frac{n\pi}{2} \).
5. Determine the values of \( \theta \) that satisfy \( 0 < \theta < \pi \) by substituting different integer values for \( n \) and checking the range.

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 these functions is crucial for solving equations involving angles, as they provide the necessary relationships to manipulate and solve for unknowns.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arctan, allow us to find angles when given a ratio. For example, if we know that an(θ) = 1, we can use the inverse tangent function to determine that θ = π/4. This concept is essential for solving trigonometric equations, as it helps to isolate the angle variable.

Periodic Nature of Trigonometric Functions

Trigonometric functions are periodic, meaning they repeat their values in regular intervals. For instance, the tangent function has a period of π, which means that an(θ) = an(θ + nπ) for any integer n. This property is important when solving equations over specific intervals, as it allows us to find all possible solutions within the given range.