Exercises 25–38 involve equations with multiple angles. Solve each equation on the interval [0, 2𝝅).3θ sec -------- = ﹣22

Verified step by step guidance

1. Recognize that the equation involves the secant function: \( \sec\left(\frac{3\theta}{2}\right) = -2 \).
2. Recall that \( \sec(x) = \frac{1}{\cos(x)} \), so \( \cos\left(\frac{3\theta}{2}\right) = -\frac{1}{2} \).
3. Determine the angles where \( \cos(x) = -\frac{1}{2} \) within the interval \([0, 2\pi)\). These angles are \( \frac{2\pi}{3} \) and \( \frac{4\pi}{3} \).
4. Set \( \frac{3\theta}{2} = \frac{2\pi}{3} + 2k\pi \) and \( \frac{3\theta}{2} = \frac{4\pi}{3} + 2k\pi \) for integer \( k \) to find all solutions.
5. Solve for \( \theta \) in each case: \( \theta = \frac{2}{3} \left(\frac{2\pi}{3} + 2k\pi\right) \) and \( \theta = \frac{2}{3} \left(\frac{4\pi}{3} + 2k\pi\right) \), ensuring solutions are within \([0, 2\pi)\).

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and secant, relate angles to ratios of sides in right triangles. Understanding these functions is crucial for solving equations involving angles, especially when dealing with multiple angles, as they can exhibit periodic behavior and specific values at key angles (e.g., 0, π/2, π, etc.).

Multiple Angle Formulas

Multiple angle formulas allow us to express trigonometric functions of multiple angles in terms of single angles. For example, the secant function can be expressed in terms of cosine, and knowing how to manipulate these formulas is essential for simplifying and solving equations that involve angles multiplied by integers, such as 3θ in this case.

Interval Notation and Solutions

Interval notation specifies the range of values for which a solution is valid. In this problem, the interval [0, 2π) indicates that we are looking for solutions within one full rotation of the unit circle. Understanding how to find and interpret solutions within this interval is key to correctly solving trigonometric equations.