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

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Recall the identity relating secant and cosine: \(\sec x = \frac{1}{\cos x}\), so \(\sec^2 x = \frac{1}{\cos^2 x}\).

Rewrite the given equation \(\sec^2 x - 2 = 0\) in terms of cosine: \(\frac{1}{\cos^2 x} - 2 = 0\).

Isolate the term with cosine: \(\frac{1}{\cos^2 x} = 2\), then take the reciprocal to get \(\cos^2 x = \frac{1}{2}\).

Take the square root of both sides to find \(\cos x = \pm \frac{1}{\sqrt{2}}\), remembering to consider both positive and negative roots.

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.

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주요 개념

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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.