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.RE.44

Capitolo 6, Problema 5.RE.44

Graph each expression and use the graph to make a conjecture, predicting what might be an identity. Then verify your conjecture algebraically.
(1 - cos 2x)/sin 2x

Guida verificata passo dopo passo

Start by understanding the given expression: \(\frac{1 - \cos 2x}{\sin 2x}\). This is a trigonometric expression involving double angles.

Graph the numerator \(1 - \cos 2x\) and the denominator \(\sin 2x\) separately over a suitable domain, such as \(x \in [0, 2\pi]\), to observe their behavior and identify points where the expression is defined.

Next, graph the entire expression \(\frac{1 - \cos 2x}{\sin 2x}\) over the same domain. Observe the shape and values of the graph to look for patterns or similarities with known trigonometric functions.

Based on the graph, make a conjecture about the identity. For example, the graph might resemble the graph of \(\tan x\) or another trigonometric function, suggesting a possible identity.

To verify the conjecture algebraically, use double-angle identities: recall that \(\cos 2x = 1 - 2\sin^2 x\) and \(\sin 2x = 2 \sin x \cos x\). Substitute these into the expression and simplify step-by-step to see if it reduces to a simpler trigonometric function.

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Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. Common identities, such as double-angle formulas, help simplify expressions and verify equivalences. Understanding these identities is essential for algebraic verification of conjectured equalities.

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Percorso guidato

5:32

Fundamental Trigonometric Identities

Graphing Trigonometric Functions

Graphing trigonometric expressions allows visualization of their behavior over intervals, revealing patterns and potential equivalences. By plotting functions like (1 - cos 2x)/sin 2x, one can observe similarities with other functions, aiding in forming conjectures about identities.

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6:04

Introduction to Trigonometric Functions

Double-Angle Formulas

Double-angle formulas express trigonometric functions of 2x in terms of x, such as cos 2x = 1 - 2sin²x and sin 2x = 2sin x cos x. These formulas are crucial for simplifying and transforming expressions involving 2x, enabling algebraic verification of identities derived from the graph.

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Percorso guidato

05:06

Double Angle Identities

Pratica correlata

Domanda del libro di testo

For each expression in Column I, use an identity to choose an expression from Column II with the same value. Choices may be used once, more than once, or not at all.

sin 300°

656

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Domanda del libro di testo

Use the given information to find cos(x - y).

sin y = - 2/3, cos x = -1/5, x in quadrant II, y in quadrant III

774

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Domanda del libro di testo

For each expression in Column I, use an identity to choose an expression from Column II with the same value. Choices may be used once, more than once, or not at all.

sin 35°

627

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Domanda del libro di testo

Work each problem.

Find the exact values of sin x, cos x, and tan x, for x = π/12 , using

a. difference identities

b. half-angle identities.

910

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Domanda del libro di testo

Use the given information to find sin(x + y).