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Ch. 5 - Trigonometric Identities
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 5 - Trigonometric IdentitiesProblema 5.52
Capitolo 6, Problema 5.52

Verify that each equation is an identity.
sin² α + tan² α + cos² α = sec² α

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Start by recalling the Pythagorean identity: \( \sin^2 \alpha + \cos^2 \alpha = 1 \).
Recall the identity for tangent and secant: \( \tan^2 \alpha = \frac{\sin^2 \alpha}{\cos^2 \alpha} \) and \( \sec^2 \alpha = 1 + \tan^2 \alpha \).
Substitute \( \tan^2 \alpha = \frac{\sin^2 \alpha}{\cos^2 \alpha} \) into the equation: \( \sin^2 \alpha + \frac{\sin^2 \alpha}{\cos^2 \alpha} + \cos^2 \alpha = \sec^2 \alpha \).
Combine the terms on the left side over a common denominator: \( \frac{\sin^2 \alpha \cdot \cos^2 \alpha + \sin^2 \alpha + \cos^4 \alpha}{\cos^2 \alpha} \).
Simplify the expression and verify if it equals \( \sec^2 \alpha \), using the identity \( \sec^2 \alpha = 1 + \tan^2 \alpha \).

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Pythagorean Identity

The Pythagorean identity states that for any angle α, the relationship sin² α + cos² α = 1 holds true. This fundamental identity is crucial in trigonometry as it connects the sine and cosine functions, allowing for simplifications and transformations in various equations.
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Pythagorean Identities

Definition of Tangent and Secant

Tangent and secant are defined in terms of sine and cosine: tan α = sin α / cos α and sec α = 1 / cos α. Understanding these definitions is essential for manipulating trigonometric identities, as they allow us to express equations in terms of sine and cosine, facilitating verification of identities.
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Graphs of Secant and Cosecant Functions

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variable where both sides are defined. Common identities include the Pythagorean identities, reciprocal identities, and quotient identities. Recognizing and applying these identities is key to verifying equations and simplifying trigonometric expressions.
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Fundamental Trigonometric Identities
Pratica correlata
Domanda del libro di testo

Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.

1 - 1/sec² x

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Verify that each equation is an identity.

(1 + sin x + cos x)² = 2(1 + sin x) (1 + cos x)

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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.

tan θ cos θ

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Write each function as an expression involving functions of θ or x alone. See Example 2.

sin (3π/4 - x)

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Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.

cot α sin α

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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.

cot² θ(1 + tan² θ)

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