In Exercises 23–26, find the exact value of each expression. Do not use a calculator.cos 2𝜋 sec 2𝜋9 9

Verified step by step guidance

1. Recognize that \( \cos 2\pi \) and \( \sec 2\pi \) are trigonometric functions evaluated at the angle \( 2\pi \).
2. Recall that \( \cos \theta = \frac{1}{\sec \theta} \), which means \( \sec \theta = \frac{1}{\cos \theta} \).
3. Evaluate \( \cos 2\pi \). Since \( 2\pi \) corresponds to a full rotation on the unit circle, \( \cos 2\pi = 1 \).
4. Use the identity \( \sec \theta = \frac{1}{\cos \theta} \) to find \( \sec 2\pi \). Since \( \cos 2\pi = 1 \), \( \sec 2\pi = \frac{1}{1} = 1 \).
5. Multiply the values: \( \cos 2\pi \cdot \sec 2\pi = 1 \cdot 1 \).

Key Concepts

Trigonometric Functions

Trigonometric functions, such as cosine (cos) and secant (sec), are fundamental in trigonometry. The cosine function relates the angle of a right triangle to the ratio of the length of the adjacent side to the hypotenuse. The secant function is the reciprocal of cosine, defined as sec(θ) = 1/cos(θ). Understanding these functions is essential for evaluating expressions involving angles.

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is a crucial tool in trigonometry for defining the values of trigonometric functions for all angles. The coordinates of points on the unit circle correspond to the cosine and sine values of the angle formed with the positive x-axis, allowing for easy evaluation of trigonometric expressions at key angles like 0, π/2, π, and 2π.

Exact Values of Trigonometric Functions

Exact values of trigonometric functions refer to the specific values of these functions at certain standard angles, such as 0, π/6, π/4, π/3, and π/2. For example, cos(0) = 1 and sec(0) = 1. Knowing these exact values allows for the simplification of expressions without the need for a calculator, which is essential in problems that require precise answers.