Exercises 39–52 involve trigonometric equations quadratic in form. Solve each equation on the interval [0, 2𝝅). sec² x - 2 = 0

Verified step by step guidance

1. Recall the identity relating secant and cosine: \(\sec x = \frac{1}{\cos x}\), so \(\sec^2 x = \frac{1}{\cos^2 x}\).
2. Rewrite the given equation \(\sec^2 x - 2 = 0\) in terms of cosine: \(\frac{1}{\cos^2 x} - 2 = 0\).
3. Isolate the term with cosine: \(\frac{1}{\cos^2 x} = 2\), then take the reciprocal to get \(\cos^2 x = \frac{1}{2}\).
4. Take the square root of both sides to find \(\cos x = \pm \frac{1}{\sqrt{2}}\), remembering to consider both positive and negative roots.
5. Determine all values of \(x\) in the interval \([0, 2\pi)\) where \(\cos x = \frac{1}{\sqrt{2}}\) and \(\cos x = -\frac{1}{\sqrt{2}}\), using the unit circle or cosine values.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. For example, the identity sec²x = 1 + tan²x allows rewriting sec²x in terms of tan²x, which is useful for solving quadratic trigonometric equations.

Quadratic Form in Trigonometric Equations

A quadratic form in trigonometric equations means the equation can be expressed as a quadratic polynomial in terms of a trigonometric function, such as tan²x or sin²x. Recognizing this form allows the use of algebraic methods like factoring or the quadratic formula to find solutions.

Solving Trigonometric Equations on a Given Interval

Solving trigonometric equations on a specific interval, such as [0, 2π), requires finding all angle solutions within that range. This involves considering the periodicity of trig functions and using inverse functions carefully to identify all valid solutions.