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

Verified step by step guidance

1. Rewrite the given equation clearly: \(\cot\left(\frac{3\theta}{2}\right) = -\sqrt{3}\).
2. Recall the definition of cotangent in terms of sine and cosine: \(\cot x = \frac{\cos x}{\sin x}\). This means we are looking for angles where the ratio of cosine to sine equals \(-\sqrt{3}\).
3. Identify the reference angle where \(\cot x = \sqrt{3}\). Since \(\cot \frac{\pi}{6} = \sqrt{3}\), the reference angle is \(\frac{\pi}{6}\).
4. Determine the quadrants where \(\cot x\) is negative. Since cotangent is positive in the first and third quadrants, it is negative in the second and fourth quadrants. So, \(\frac{3\theta}{2}\) lies in the second or fourth quadrant.
5. Write the general solutions for \(\frac{3\theta}{2}\) using the reference angle \(\frac{\pi}{6}\) in the second and fourth quadrants: \(\frac{3\theta}{2} = \pi - \frac{\pi}{6} + 2k\pi\) and \(\frac{3\theta}{2} = 2\pi - \frac{\pi}{6} + 2k\pi\), where \(k\) is any integer. Then solve for \(\theta\) by multiplying both sides by \(\frac{2}{3}\).

Key Concepts

Multiple-Angle Trigonometric Equations

These equations involve trigonometric functions with angles that are multiples of the variable, such as 3θ. Solving them requires isolating the trigonometric function and then finding all angle solutions within the given interval, considering the periodicity of the function.

Cotangent Function and Its Properties

Cotangent is the reciprocal of the tangent function, defined as cot(θ) = cos(θ)/sin(θ). Understanding its values, periodicity (π), and behavior is essential for solving equations involving cotangent, especially when equated to specific constants like -√3.

Solving Trigonometric Equations on a Restricted Interval

When solving trigonometric equations on [0, 2π), it is important to find all solutions within this domain. For multiple-angle equations, solutions for the inner angle (e.g., 3θ) must be found first, then adjusted to the original variable's interval by dividing and considering all valid solutions.