In Exercises 8–13, find the exact value of each expression. Do not use a calculator.sec 22𝜋3

Verified step by step guidance

1. Recognize that \( \sec \theta = \frac{1}{\cos \theta} \). Therefore, we need to find \( \cos \left( \frac{22\pi}{3} \right) \).
2. Convert \( \frac{22\pi}{3} \) into an angle within the standard range \([0, 2\pi)\) by subtracting multiples of \(2\pi\).
3. Calculate \( \frac{22\pi}{3} - 2\pi \times n \) where \( n \) is the largest integer such that the result is non-negative and less than \( 2\pi \).
4. Determine the reference angle for the resulting angle and identify the quadrant in which it lies.
5. Use the reference angle and the quadrant to find \( \cos \theta \), then calculate \( \sec \theta = \frac{1}{\cos \theta} \).

Key Concepts

Secant Function

The secant function, denoted as sec(θ), is the reciprocal of the cosine function. It is defined as sec(θ) = 1/cos(θ). Understanding secant is crucial for solving problems involving angles, especially in trigonometric identities and equations.

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is fundamental in trigonometry as it provides a geometric interpretation of the sine, cosine, and tangent functions. Angles measured in radians correspond to points on the unit circle, which helps in determining the values of trigonometric functions for various angles.

Angle Measurement in Radians

Radians are a unit of angular measure where one radian is the angle subtended at the center of a circle by an arc equal in length to the radius. In this problem, the angle 22π/3 can be simplified by converting it to a more manageable angle within the standard range of 0 to 2π. Understanding how to convert and interpret angles in radians is essential for evaluating trigonometric functions accurately.