Ch. 1 - Angles and the Trigonometric Functions

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 1 - Angles and the Trigonometric Functions

Problem 1.RE.51

Chapter 1, Problem 1.RE.51

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

Verified step by step guidance

Recall the definition of secant: \(\sec \theta = \frac{1}{\cos \theta}\). So, to find \(\sec \frac{7\pi}{4}\), we first need to find \(\cos \frac{7\pi}{4}\).

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 \(2\pi\): \(2\pi - \frac{7\pi}{4} = \frac{\pi}{4}\).

Determine the cosine value of the reference angle \(\frac{\pi}{4}\). We know \(\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}\).

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 \(\cos \frac{7\pi}{4} = \frac{\sqrt{2}}{2}\).

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.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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