In Exercises 49–59, find the exact value of each expression. Do not use a calculator. sec 7𝜋 / 4

Verified step by step guidance

1. Recall the definition of secant: \(\sec \theta = \frac{1}{\cos \theta}\). So, to find \(\sec \frac{7\pi}{4}\), we first need to find \(\cos \frac{7\pi}{4}\).
2. Identify the angle \(\frac{7\pi}{4}\) on the unit circle. Since \(\frac{7\pi}{4}\) is between \(2\pi\) and \(\frac{3\pi}{2}\), find its reference angle by subtracting from \(2\pi\): \(2\pi - \frac{7\pi}{4} = \frac{\pi}{4}\).
3. Determine the cosine value of the reference angle \(\frac{\pi}{4}\). We know \(\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}\).
4. Determine the sign of cosine in the quadrant where \(\frac{7\pi}{4}\) lies. Since \(\frac{7\pi}{4}\) is in the fourth quadrant, cosine is positive there, so \(\cos \frac{7\pi}{4} = \frac{\sqrt{2}}{2}\).
5. Finally, use the secant definition to write \(\sec \frac{7\pi}{4} = \frac{1}{\cos \frac{7\pi}{4}} = \frac{1}{\frac{\sqrt{2}}{2}}\). Simplify this expression to get 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(7π/4), you first find cos(7π/4) and then take its reciprocal. This relationship is fundamental for evaluating secant values without a calculator.

Reference Angles and Unit Circle Values

Angles like 7π/4 can be related to reference angles within the first quadrant to determine their trigonometric values. Since 7π/4 is in the fourth quadrant, its reference angle is π/4. Knowing the cosine value at π/4 and the sign of cosine in the fourth quadrant helps find cos(7π/4) exactly.

Sign of Trigonometric Functions in Quadrants

The sign of cosine and secant depends on the quadrant of the angle. In the fourth quadrant (where 7π/4 lies), cosine is positive, so secant will also be positive. Understanding the ASTC (All Students Take Calculus) rule helps determine the correct sign of the trigonometric value.