Skip to main content
Ch. 1 - Angles and the Trigonometric Functions
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 3rd EditionCh. 1 - Angles and the Trigonometric FunctionsProblema 1.RE.51
Capitolo 1, Problema 1.RE.51

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

Guida verificata passo dopo passo
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 \(\cos \frac{7\pi}{4}\).
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.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
5m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

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.
Video consigliato:
Percorso guidato
6:22
Graphs of Secant and Cosecant Functions

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.
Video consigliato:
Percorso guidato
5:31
Reference Angles on the Unit Circle

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.
Video consigliato:
Percorso guidato
6:36
Quadratic Formula
Pratica correlata
Domanda del libro di testo

In Exercises 49–59, find the exact value of each expression. Do not use a calculator. cos (11𝜋 / 6)

707
views
Domanda del libro di testo

In Exercises 49–59, find the exact value of each expression. Do not use a calculator. cos(-35𝜋 / 6)

635
views
Domanda del libro di testo

In Exercises 49–59, find the exact value of each expression. Do not use a calculator. sin 240°

729
views
Domanda del libro di testo

In Exercises 49–59, find the exact value of each expression. Do not use a calculator. sin(-𝜋/3)

697
views
Domanda del libro di testo

In Exercises 41–56, use the circle shown in the rectangular coordinate system to draw each angle in standard position. State the quadrant in which the angle lies. When an angle's measure is given in radians, work the exercise without converting to degrees.

420°

625
views
Domanda del libro di testo

In Exercises 1–6, the measure of an angle is given. Classify the angle as acute, right, obtuse, or straight. 135°

807
views