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

Verified step by step guidance

1. Rewrite the given equation \(3\theta \sec \frac{3\theta}{2} = -2\) to isolate the secant function: \(\sec \frac{3\theta}{2} = -2\).
2. Recall that \(\sec x = \frac{1}{\cos x}\), so rewrite the equation as \(\frac{1}{\cos \frac{3\theta}{2}} = -2\).
3. Invert both sides to express in terms of cosine: \(\cos \frac{3\theta}{2} = -\frac{1}{2}\).
4. Find all angles \(\alpha = \frac{3\theta}{2}\) in the interval \([0, 3\pi)\) (since \(\theta \in [0, 2\pi)\), multiplying by \(\frac{3}{2}\) extends the interval) where \(\cos \alpha = -\frac{1}{2}\).
5. Solve for \(\theta\) by isolating it: \(\theta = \frac{2}{3} \alpha\), and then select all solutions for \(\theta\) that lie within \([0, 2\pi)\).

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 understanding how to manipulate and simplify expressions with these multiple angles to find all possible solutions within a given interval.

Secant Function and Its Properties

The secant function, sec(θ), is the reciprocal of cosine, defined as sec(θ) = 1/cos(θ). Understanding its domain, range, and behavior is essential, especially since sec(θ) can be undefined where cosine is zero, and it can take values less than -1 or greater than 1.

Solving Trigonometric Equations on a Restricted Interval

When solving equations on [0, 2π), it is important to find all solutions within one full rotation of the unit circle. This involves considering the periodicity of the trigonometric functions and adjusting solutions for multiple angles accordingly.