Ch. 1 - Angles and the Trigonometric Functions

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Blitzer 3rd Edition

Ch. 1 - Angles and the Trigonometric Functions

Problema 1.3.64

Capitolo 1, Problema 1.3.64

In Exercises 61–86, use reference angles to find the exact value of each expression. Do not use a calculator. sec 240°

Guida verificata passo dopo passo

Identify the reference angle for 240°. Since 240° is in the third quadrant, subtract 180° from 240° to find the reference angle: 240° - 180° = 60°.

Recall that the secant function, \( \sec \theta \), is the reciprocal of the cosine function, \( \cos \theta \). Therefore, \( \sec \theta = \frac{1}{\cos \theta} \).

Determine the sign of the secant function in the third quadrant. In the third quadrant, cosine is negative, so secant will also be negative.

Use the reference angle to find \( \cos 60° \). From trigonometric tables or knowledge, \( \cos 60° = \frac{1}{2} \).

Calculate \( \sec 240° \) using the reference angle: \( \sec 240° = \frac{1}{\cos 240°} = \frac{1}{-\frac{1}{2}} = -2 \).

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Reference Angles

A reference angle is the acute angle formed between the terminal side of a given angle and the x-axis. It helps simplify trigonometric calculations by relating angles in different quadrants to their acute counterparts, allowing the use of known values for sine, cosine, and other functions.

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Percorso guidato

5:31

Reference Angles on the Unit Circle

Secant Function and Its Relationship to Cosine

The secant function, sec(θ), is the reciprocal of the cosine function, defined as sec(θ) = 1/cos(θ). Understanding this relationship allows you to find secant values by first determining the cosine of the angle, especially useful when working with exact values.

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Percorso guidato

6:22

Graphs of Secant and Cosecant Functions

Trigonometric Values in Different Quadrants

The sign of trigonometric functions depends on the quadrant in which the angle lies. For 240°, located in the third quadrant, cosine (and thus secant) is negative. Recognizing quadrant signs is essential to assign the correct sign to the exact trigonometric value.

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Percorso guidato

6:36

Quadratic Formula

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.

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

sec 3𝜋/2

450

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Domanda del libro di testo

In Exercises 57–70, find a positive angle less than or that is coterminal with the given angle. - 38𝜋/9

668

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Domanda del libro di testo

In Exercises 69–70, express the exact value of each function as a single fraction. Do not use a calculator. If f(θ) = 2 cos θ - cos 2θ, find f(𝜋/6).

525