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 5.36

Capitolo 6, Problema 5.36

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

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Recognize that \( \cot t \) is the cotangent of \( t \), which is defined as \( \frac{1}{\tan t} \).

Substitute \( \cot t \) with \( \frac{1}{\tan t} \) in the expression \( \cot t \tan t \).

The expression becomes \( \frac{1}{\tan t} \times \tan t \).

Apply the property of multiplication that states \( a \times \frac{1}{a} = 1 \) for any non-zero \( a \).

Conclude that \( \frac{1}{\tan t} \times \tan t = 1 \), simplifying the expression to a constant.

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Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Fundamental identities, such as the Pythagorean identities, reciprocal identities, and quotient identities, serve as the foundation for simplifying trigonometric expressions. Understanding these identities is crucial for manipulating and simplifying expressions like 'cot t tan t'.

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Percorso guidato

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Fundamental Trigonometric Identities

Cotangent and Tangent Functions

The cotangent (cot) and tangent (tan) functions are fundamental trigonometric functions defined as the ratios of the sides of a right triangle. Specifically, cotangent is the reciprocal of tangent, expressed as cot t = 1/tan t. Recognizing the relationship between these functions is essential for simplifying expressions involving them, such as 'cot t tan t', which simplifies to 1.

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Percorso guidato

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Introduction to Cotangent Graph

Simplification of Trigonometric Expressions

Simplifying trigonometric expressions involves using identities and algebraic manipulation to reduce complex expressions to simpler forms. This process often includes factoring, combining like terms, and substituting equivalent expressions. In the case of 'cot t tan t', applying the identity that cot t is the reciprocal of tan t leads to a straightforward simplification to 1.

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Percorso guidato

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Simplifying Trig Expressions

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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.

1/ tan² α + cot α tan α

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Factor each trigonometric expression.

sec² θ - 1

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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.

-sec² (-θ) + sin² (-θ) + cos² (-θ)

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Match each expression in Column I with its value in Column II.

10. cos 67.5°

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Verify that each equation is an identity.

sec⁴ x - sec² x = tan⁴ x + tan² x

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