Solve each equation in x over the interval [0, 2π) and each equation in θ over the interval [0°, 360°). Give exact solutions.
8 sec² x/2 = 4

Verified step by step guidance

1. Start with the given equation: \(8 \sec^{2} \frac{x}{2} = 4\).
2. Divide both sides of the equation by 8 to isolate \(\sec^{2} \frac{x}{2}\): \(\sec^{2} \frac{x}{2} = \frac{4}{8} = \frac{1}{2}\).
3. Recall the identity \(\sec \theta = \frac{1}{\cos \theta}\), so \(\sec^{2} \theta = \frac{1}{\cos^{2} \theta}\). Substitute this into the equation to get \(\frac{1}{\cos^{2} \frac{x}{2}} = \frac{1}{2}\).
4. Invert both sides to solve for \(\cos^{2} \frac{x}{2}\): \(\cos^{2} \frac{x}{2} = 2\).
5. Analyze the equation \(\cos^{2} \frac{x}{2} = 2\) and determine if there are any real solutions for \(x\) in the interval \([0, 2\pi)\), considering the range of the cosine function.

Key Concepts

Trigonometric Functions and Their Definitions

Understanding the secant function (sec) is essential; it is the reciprocal of the cosine function, so sec(θ) = 1/cos(θ). This relationship allows converting secant equations into cosine equations, which are often easier to solve.

Solving Trigonometric Equations

To solve equations like 8 sec²(x/2) = 4, first isolate the trigonometric function, then use algebraic manipulation to find the values of the angle. Recognizing how to handle squared trigonometric functions and applying inverse functions is key.

Interval and Domain Considerations for Solutions

Solutions must be found within specified intervals, such as [0, 2π) for radians or [0°, 360°) for degrees. Understanding how to find all solutions within these intervals, including using periodicity and symmetry of trig functions, ensures complete and exact answers.