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 11
Capitolo 1, Problema 11

In Exercises 8–13, find the exact value of each expression. Do not use a calculator. sec 22𝜋 3

Guida verificata passo dopo passo
1
Recognize that the expression is \(\sec \left( \frac{22\pi}{3} \right)\), which involves the secant function of an angle measured in radians.
Recall that the secant function is the reciprocal of the cosine function, so \(\sec \theta = \frac{1}{\cos \theta}\). Therefore, finding \(\sec \left( \frac{22\pi}{3} \right)\) is equivalent to finding \(\frac{1}{\cos \left( \frac{22\pi}{3} \right)}\).
Since the cosine function is periodic with period \(2\pi\), reduce the angle \(\frac{22\pi}{3}\) by subtracting multiples of \(2\pi\) until the angle lies within the standard interval \([0, 2\pi)\): calculate \(\frac{22\pi}{3} - 2\pi \times k\) for an integer \(k\) such that the result is between \(0\) and \(2\pi\).
Once the angle is reduced to an equivalent angle \(\theta_{reduced}\) in \([0, 2\pi)\), evaluate \(\cos \theta_{reduced}\) using known values or unit circle properties.
Finally, compute \(\sec \left( \frac{22\pi}{3} \right) = \frac{1}{\cos \theta_{reduced}}\) to find 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:
6m

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(θ), you first determine cos(θ) and then take its reciprocal. This relationship is fundamental when evaluating trigonometric expressions without a calculator.
Video consigliato:
Percorso guidato
6:22
Graphs of Secant and Cosecant Functions

Evaluating Trigonometric Functions at Special Angles

Angles like 2π/3 are special angles on the unit circle with known sine and cosine values. Recognizing these angles allows you to find exact trigonometric values using the unit circle, avoiding decimal approximations. For 2π/3, cosine is negative one-half, which is key to finding sec(2π/3).
Video consigliato:
Percorso guidato
3:48
Evaluate Composite Functions - Special Cases

Using the Unit Circle for Exact Values

The unit circle provides exact values for sine and cosine at various angles measured in radians. By locating the angle 2π/3 on the unit circle, you can identify the coordinates (cosine, sine) and thus find exact trigonometric values. This method is essential for solving problems without calculators.
Video consigliato:
Percorso guidato
06:11
Introduction to the Unit Circle
Pratica correlata
Domanda del libro di testo

In Exercises 5–18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of 0, 𝜋, 𝜋, 𝜋, 2𝜋, 5𝜋, 𝜋, 7𝜋, 4𝜋, 3𝜋, 5𝜋, 11𝜋, and 2𝜋. 6 3 2 3 6 6 3 2 3 6 Use the (x,y) coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined.

tan 0

1138
views
Domanda del libro di testo

In Exercises 8–12, draw each angle in standard position. 8𝜋 3

762
views
Domanda del libro di testo

In Exercises 5–18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of


0, 𝜋, 𝜋, 𝜋, 2𝜋, 5𝜋, 𝜋, 7𝜋, 4𝜋, 3𝜋, 5𝜋, 11𝜋, and 2𝜋.

6 3 2 3 6 6 3 2 3 6


Use the (x,y) coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined.

<IMAGE>


In Exercises 11–18, continue to refer to the figure at the bottom of the previous page.

csc 7𝜋/6

414
views
Domanda del libro di testo

In Exercises 8–12, draw each angle in standard position. -135°

798
views
Domanda del libro di testo

Use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.


<IMAGE>


sec 45°

550
views
Domanda del libro di testo

In Exercises 8–13, find the exact value of each expression. Do not use a calculator. cot (-8𝜋/3)

669
views