Exercises 25–38 involve equations with multiple angles. Solve each equation on the interval [0, 2𝝅). tan(x/2) = √3

Verified step by step guidance

1. Rewrite the equation clearly: \( \tan\left(\frac{x}{2}\right) = \sqrt{3} \).
2. Recall that \( \tan(\theta) = \sqrt{3} \) at specific standard angles. Identify the general solutions for \( \theta \) where \( \tan(\theta) = \sqrt{3} \).
3. Since \( \theta = \frac{x}{2} \), express the solutions for \( x \) by multiplying the general solutions for \( \theta \) by 2.
4. Use the periodicity of the tangent function, which has period \( \pi \), to write the general solution for \( \theta \) as \( \theta = \frac{\pi}{3} + k\pi \), where \( k \) is any integer.
5. Apply the interval restriction \( x \in [0, 2\pi) \) to find all valid values of \( x \) by substituting \( k \) values and checking which \( x \) values fall within the interval.

Key Concepts

Solving Trigonometric Equations

Solving trigonometric equations involves finding all angle values within a given interval that satisfy the equation. This often requires isolating the trigonometric function and using known values or identities to determine possible solutions.

Tangent Function and Its Values

The tangent function, tan(θ), is periodic with period π and relates the ratio of sine to cosine. Knowing key tangent values, such as tan(π/3) = √3, helps identify solutions to equations involving tangent.

Multiple-Angle Equations and Interval Restrictions

When the angle inside the trigonometric function is a multiple of the variable (e.g., tan(x/2)), solutions must be found by considering the function's period and the specified interval, ensuring all valid solutions within [0, 2π) are included.