Ch. 5 - Trigonometric Identities

Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo

Lial 12th Edition

Ch. 5 - Trigonometric Identities

Problema 5.6.8

Capitolo 6, Problema 5.6.8

Match each expression in Column I with its value in Column II.
8. tan (-π/8)

Guida verificata passo dopo passo

Recall the definition of the tangent function: \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\). This will help understand the behavior of \(\tan(-\frac{\pi}{8})\).

Use the odd function property of tangent: \(\tan(-\theta) = -\tan(\theta)\). This means \(\tan(-\frac{\pi}{8}) = -\tan(\frac{\pi}{8})\).

Focus on finding \(\tan(\frac{\pi}{8})\). Recognize that \(\frac{\pi}{8}\) is \(22.5^\circ\), which is half of \(45^\circ\).

Apply the half-angle formula for tangent: \(\tan\left(\frac{\alpha}{2}\right) = \frac{1 - \cos(\alpha)}{\sin(\alpha)}\) or alternatively \(\tan\left(\frac{\alpha}{2}\right) = \sqrt{\frac{1 - \cos(\alpha)}{1 + \cos(\alpha)}}\) depending on the quadrant.

Calculate \(\tan(\frac{\pi}{8})\) using the half-angle formula with \(\alpha = \frac{\pi}{4}\), then use the odd function property to find \(\tan(-\frac{\pi}{8})\) by negating the result.

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Tangent Function and Its Properties

The tangent function, tan(θ), is defined as the ratio of the sine and cosine of an angle θ. It is periodic with period π and is odd, meaning tan(-θ) = -tan(θ). Understanding these properties helps evaluate tangent values for negative angles.

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Percorso guidato

5:43

Introduction to Tangent Graph

Reference Angles and Angle Reduction

Reference angles are acute angles used to find the trigonometric values of angles outside the first quadrant. For negative angles like -π/8, the value of tan(-π/8) can be found by using the odd property of tangent and the positive reference angle π/8.

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Percorso guidato

5:31

Reference Angles on the Unit Circle

Exact Values of Special Angles

Certain angles, such as π/8 (22.5°), have known exact trigonometric values expressed in surds or radicals. Knowing or deriving the exact value of tan(π/8) allows precise calculation of tan(-π/8) without a calculator.

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Percorso guidato

04:39

45-45-90 Triangles

Pratica correlata

Domanda del libro di testo

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.

1 + cot(-θ)/cot(-θ)

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Domanda del libro di testo

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.

csc θ - sin θ

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Domanda del libro di testo

Verify that each equation is an identity.

(1 - cos θ)/(1 + cos θ) = 2 csc² θ - 2 csc θ cot θ - 1

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Domanda del libro di testo

Identify the basic trigonometric function graphed, and determine whether it is even or odd.

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