Find each exact function value. See Example 2.
sec 23π/6

Verified step by step guidance

1. First, recognize that the angle given is \( \frac{23\pi}{6} \), which is an improper fraction greater than \( 2\pi \). To simplify, subtract multiples of \( 2\pi \) to find a coterminal angle within the interval \( [0, 2\pi) \). Since \( 2\pi = \frac{12\pi}{6} \), subtract \( 12\pi/6 \) twice: \( \frac{23\pi}{6} - 2 \times \frac{12\pi}{6} = \frac{23\pi}{6} - \frac{24\pi}{6} = -\frac{\pi}{6} \).
2. Because \( -\frac{\pi}{6} \) is negative, add \( 2\pi \) once to get a positive coterminal angle: \( -\frac{\pi}{6} + 2\pi = -\frac{\pi}{6} + \frac{12\pi}{6} = \frac{11\pi}{6} \). So, \( \sec \frac{23\pi}{6} = \sec \frac{11\pi}{6} \).
3. Recall that \( \sec \theta = \frac{1}{\cos \theta} \). Therefore, to find \( \sec \frac{11\pi}{6} \), first find \( \cos \frac{11\pi}{6} \).
4. Use the unit circle or cosine properties: \( \frac{11\pi}{6} \) is in the fourth quadrant, where cosine is positive. The reference angle is \( 2\pi - \frac{11\pi}{6} = \frac{\pi}{6} \). So, \( \cos \frac{11\pi}{6} = \cos \frac{\pi}{6} \).
5. Recall the exact value \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \). Therefore, \( \sec \frac{11\pi}{6} = \frac{1}{\cos \frac{11\pi}{6}} = \frac{1}{\frac{\sqrt{3}}{2}} \).

Key Concepts

Understanding the Secant Function

The secant function, sec(θ), is the reciprocal of the cosine function, defined as sec(θ) = 1/cos(θ). To find sec(θ), you first determine the cosine of the angle θ and then take its reciprocal. This relationship is fundamental when evaluating secant values exactly.

Angle Reduction Using Coterminal Angles

Angles greater than 2π or less than 0 can be simplified by subtracting or adding multiples of 2π to find a coterminal angle within the standard interval [0, 2π). For 23π/6, subtracting 2π (12π/6) twice reduces it to an equivalent angle, making it easier to evaluate trigonometric functions.

Exact Values of Trigonometric Functions for Special Angles

Certain angles, such as π/6, π/4, and π/3, have well-known exact trigonometric values involving square roots and fractions. Recognizing that 23π/6 reduces to a special angle allows you to use these exact values rather than decimal approximations, ensuring precise answers.