In Exercises 8–13, find the exact value of each expression. Do not use a calculator. sec 22𝜋 3

Verified step by step guidance

1. Recognize that the expression is \(\sec \left( \frac{22\pi}{3} \right)\), which involves the secant function of an angle measured in radians.
2. Recall that the secant function is the reciprocal of the cosine function, so \(\sec \theta = \frac{1}{\cos \theta}\). Therefore, finding \(\sec \left( \frac{22\pi}{3} \right)\) is equivalent to finding \(\frac{1}{\cos \left( \frac{22\pi}{3} \right)}\).
3. Since the cosine function is periodic with period \(2\pi\), reduce the angle \(\frac{22\pi}{3}\) by subtracting multiples of \(2\pi\) until the angle lies within the standard interval \([0, 2\pi)\): calculate \(\frac{22\pi}{3} - 2\pi \times k\) for an integer \(k\) such that the result is between \(0\) and \(2\pi\).
4. Once the angle is reduced to an equivalent angle \(\theta_{reduced}\) in \([0, 2\pi)\), evaluate \(\cos \theta_{reduced}\) using known values or unit circle properties.
5. Finally, compute \(\sec \left( \frac{22\pi}{3} \right) = \frac{1}{\cos \theta_{reduced}}\) to find the exact value.

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 cos(θ) and then take its reciprocal. This relationship is fundamental when evaluating trigonometric expressions without a calculator.

Evaluating Trigonometric Functions at Special Angles

Angles like 2π/3 are special angles on the unit circle with known sine and cosine values. Recognizing these angles allows you to find exact trigonometric values using the unit circle, avoiding decimal approximations. For 2π/3, cosine is negative one-half, which is key to finding sec(2π/3).

Using the Unit Circle for Exact Values

The unit circle provides exact values for sine and cosine at various angles measured in radians. By locating the angle 2π/3 on the unit circle, you can identify the coordinates (cosine, sine) and thus find exact trigonometric values. This method is essential for solving problems without calculators.