Question

Use the fact that the trigonometric functions are periodic to find the exact value of each expression. Do not use a calculator. $$ \sec \frac{25 \pi}{6} $$

Short answer

\( \frac{2 \sqrt{3}}{3} \)

Step-by-step solution

- Understand the Periodicity of the Secant Function

Recall that the secant function is related to the cosine function: \ \(\sec(x) = \frac{1}{\cos(x)} \). The cosine function is periodic with a period of \(2\pi\), which means \(\cos(x) = \cos(x + 2k\pi)\) for any integer \(k\). This property applies to the secant function as well.

- Reduce the Angle

To find \(\sec \frac{25 \pi}{6}\), reduce \( \frac{25 \pi}{6} \) by subtracting multiples of \(2\pi\): \ \( \frac{25 \pi}{6} - 4 \pi = \frac{25 \pi}{6} - \frac{24 \pi}{6} = \frac{\pi}{6} \). So, \( \sec \frac{25 \pi}{6} = \sec \frac{\pi}{6} \).

- Find the Secant of the Reduced Angle

Next, find the value of \( \sec \frac{\pi}{6} \). Knowing that \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \), we get: \ \( \sec \frac{\pi}{6} = \frac{1}{\cos \frac{\pi}{6}} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} = \frac{2 \sqrt{3}}{3} \).

Key concepts

Periodicity

Trigonometric functions, including the secant function, exhibit periodic behavior. This means their values repeat at regular intervals. For the cosine function, the period is \(2\pi\). This translates to the secant function, since \( \sec(x) = \frac{1}{\cos(x)} \). Thus, \( \sec(x) = \sec(x + 2k\pi) \) for any integer \(k\). This property allows us to reduce angles within a limited interval, simplifying the computation of trigonometric values. By understanding and applying periodicity, we make calculations easier and often avoid unnecessary complexities.

Secant Function

The secant function, represented by \( \sec(x) \), is the reciprocal of the cosine function: \( \sec(x) = \frac{1}{\cos(x)} \). It is undefined when the cosine of \(x\) is zero since division by zero is not defined. To find secant values, we first determine the cosine value and then compute its reciprocal. For instance, given \( \sec \frac{\pi}{6} \), we find \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \), and thus, \( \sec \frac{\pi}{6} = \frac{2}{\sqrt{3}} \). This simplifies to \( \frac{2 \sqrt{3}}{3} \).

Angle Reduction

Angle reduction involves simplifying a given angle by subtracting multiples of \(2\pi\) to fall within a standard interval, typically \([0, 2\pi)\). For example, to reduce \( \frac{25 \pi}{6} \), we subtract \(4 \pi\) (which is equivalent to \( \frac{24 \pi}{6} \)), giving us \( \frac{\pi}{6} \). This is because \( \frac{25 \pi}{6} - \frac{24 \pi}{6} = \frac{\pi}{6} \). Reduced angles help in finding trigonometric values more simply by converting them to familiar, standard angles.

Trigonometric Identities

Trigonometric identities are fundamental relationships between trigonometric functions. For example, \( \sec(x) = \frac{1}{\cos(x)} \) links secant and cosine. Identities like these simplify complex expressions and solve trigonometric equations. Other essential identities include \( \cos^2(x) + \sin^2(x) = 1 \) and \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). Memorizing these identities is incredibly helpful in trigonometry, providing tools to manipulate and simplify expressions efficiently.

Cosine Function

The cosine function \( \cos(x) \) is a fundamental trigonometric function representing the x-coordinate of a point on the unit circle. It is periodic with a period of \(2\pi\), meaning \( \cos(x) = \cos(x + 2k\pi) \) for any integer \(k\). Values of the cosine function range between -1 and 1. Key angle values to remember include \( \cos(0) = 1 \), \( \cos(\pi/2) = 0 \), \( \cos(\pi) = -1 \), and \( \cos(3\pi/2) = 0 \). For example, \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \), which we used in our problem to find \( \sec \frac{\pi}{6} \).