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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 5.38
Capitolo 6, Problema 5.38

Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.
(csc θ sec θ)/cot θ

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Start by recalling the fundamental trigonometric identities: \( \csc \theta = \frac{1}{\sin \theta} \), \( \sec \theta = \frac{1}{\cos \theta} \), and \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).
Substitute these identities into the expression \( \frac{\csc \theta \sec \theta}{\cot \theta} \) to get \( \frac{\frac{1}{\sin \theta} \cdot \frac{1}{\cos \theta}}{\frac{\cos \theta}{\sin \theta}} \).
Simplify the expression by multiplying the numerators and denominators: \( \frac{\frac{1}{\sin \theta \cos \theta}}{\frac{\cos \theta}{\sin \theta}} \).
To simplify further, multiply by the reciprocal of the denominator: \( \frac{1}{\sin \theta \cos \theta} \times \frac{\sin \theta}{\cos \theta} \).
Cancel out the common terms in the numerator and the denominator to simplify the expression to a single trigonometric function.

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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 (csc θ sec θ)/cot θ.
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Fundamental Trigonometric Identities

Reciprocal Functions

Reciprocal functions in trigonometry refer to pairs of functions that are inverses of each other. For example, cosecant (csc) is the reciprocal of sine (sin), secant (sec) is the reciprocal of cosine (cos), and cotangent (cot) is the reciprocal of tangent (tan). Recognizing these relationships allows for easier simplification of expressions by substituting one function for its reciprocal.
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Secant, Cosecant, & Cotangent on the Unit Circle

Quotient Identities

Quotient identities express the relationships between the sine, cosine, and tangent functions. Specifically, tangent is defined as the ratio of sine to cosine (tan θ = sin θ/cos θ), and cotangent is the reciprocal of tangent (cot θ = cos θ/sin θ). Utilizing these identities is essential for simplifying expressions involving cotangent, as seen in the given expression (csc θ sec θ)/cot θ.
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Quotients of Complex Numbers in Polar Form
Pratica correlata
Domanda del libro di testo

Perform each transformation. See Example 2.

Write sec x in terms of sin 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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Find sinθ.

sec θ = 7/2, tan θ < 0

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Verify that each equation is an identity.

sin θ + cos θ = sin θ/(1 - cot θ) + cos θ/(1 - tan θ)

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Verify that each equation is an identity.

tan α/sec 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.

tan θ cos θ

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