In Exercises 49–59, find the exact value of each expression. Do not use a calculator.sec 7𝜋4

Verified step by step guidance

1. Recognize that \( \sec \theta = \frac{1}{\cos \theta} \). Therefore, we need to find \( \cos \left( \frac{7\pi}{4} \right) \).
2. Convert \( \frac{7\pi}{4} \) into degrees if necessary, or recognize it as an angle in radians. \( \frac{7\pi}{4} \) is equivalent to \( 315^\circ \).
3. Determine the reference angle for \( 315^\circ \) or \( \frac{7\pi}{4} \). The reference angle is \( 360^\circ - 315^\circ = 45^\circ \) or \( 2\pi - \frac{7\pi}{4} = \frac{\pi}{4} \).
4. Identify the quadrant in which \( \frac{7\pi}{4} \) lies. It is in the fourth quadrant, where cosine is positive.
5. Use the reference angle to find \( \cos \left( \frac{7\pi}{4} \right) \). Since the reference angle is \( \frac{\pi}{4} \), \( \cos \left( \frac{7\pi}{4} \right) = \cos \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} \). Therefore, \( \sec \left( \frac{7\pi}{4} \right) = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2} \).

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 the secant function is crucial for evaluating expressions involving angles, particularly 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 those 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 context, 7π/4 radians corresponds to an angle that can be located on the unit circle. Understanding how to convert between degrees and radians is essential for evaluating trigonometric functions accurately.