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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

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


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sec 45°

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1
Recall the definition of secant in terms of cosine: \(\sec \theta = \frac{1}{\cos \theta}\).
Identify the angle given: \(45^\circ\).
Find the value of \(\cos 45^\circ\). From the special right triangle (45°-45°-90°), \(\cos 45^\circ = \frac{\sqrt{2}}{2}\).
Substitute this value into the secant formula: \(\sec 45^\circ = \frac{1}{\frac{\sqrt{2}}{2}}\).
Rationalize the denominator by multiplying numerator and denominator by \(\sqrt{2}\) to eliminate the square root in the denominator.

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Definition of Secant Function

The secant function, sec(θ), is the reciprocal of the cosine function, defined as sec(θ) = 1/cos(θ). It relates the hypotenuse to the adjacent side in a right triangle and is used to find ratios involving angles.
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Percorso guidato
6:22
Graphs of Secant and Cosecant Functions

Exact Values of Trigonometric Functions for Special Angles

Certain angles like 45° have well-known exact trigonometric values. For 45°, cos(45°) = √2/2, so sec(45°) = 1/(√2/2) = √2. Knowing these values helps evaluate expressions without a calculator.
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Percorso guidato
6:04
Introduction to Trigonometric Functions

Rationalizing the Denominator

Rationalizing the denominator involves eliminating square roots from the denominator of a fraction by multiplying numerator and denominator by a suitable radical. This simplifies expressions and is often required for final answers.
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2:58
Rationalizing Denominators
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

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Domanda del libro di testo

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

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Domanda del libro di testo

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

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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.

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

csc 7𝜋/6

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Domanda del libro di testo

In Exercises 9–16, evaluate the trigonometric function at the quadrantal angle, or state that the expression is undefined. tan 𝜋

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Domanda del libro di testo

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

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