In Exercises 127–130, solve each equation on the interval [0, 2𝝅) by first rewriting the equation in terms of sines or cosines. sec² x + 3 sec x + 2 = 0

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Recall that secant is the reciprocal of cosine, so rewrite the equation in terms of cosine: replace \( \sec x \) with \( \frac{1}{\cos x} \). The equation becomes \( \left( \frac{1}{\cos x} \right)^2 + 3 \left( \frac{1}{\cos x} \right) + 2 = 0 \).

Multiply through the entire equation by \( \cos^2 x \) to clear the denominators, resulting in a quadratic equation in terms of \( \cos x \): \( 1 + 3 \cos x + 2 \cos^2 x = 0 \).

Rewrite the quadratic equation in standard form: \( 2 \cos^2 x + 3 \cos x + 1 = 0 \).

Use the quadratic formula or factoring to solve for \( \cos x \). The quadratic formula is \( \cos x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a=2 \), \( b=3 \), and \( c=1 \).

Find all values of \( x \) in the interval \( [0, 2\pi) \) such that \( \cos x \) equals the solutions found. Use the unit circle or inverse cosine function to determine these angles.

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주요 개념

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Secant Function and Its Relationship to Cosine

The secant function, sec x, is the reciprocal of the cosine function, defined as sec x = 1/cos x. Understanding this relationship allows rewriting equations involving sec x in terms of cosine, which is often easier to solve.

Solving Quadratic Equations in Trigonometric Functions

Trigonometric equations like sec² x + 3 sec x + 2 = 0 can be treated as quadratic equations by substituting a trigonometric expression (e.g., sec x = y). Solving the quadratic yields values for the trigonometric function, which can then be used to find the angle solutions.

Finding Solutions on a Specific Interval [0, 2π)

When solving trigonometric equations, it is important to find all solutions within the given interval, here [0, 2π). This involves considering the periodicity of sine and cosine functions and checking all possible angles that satisfy the equation within one full rotation.