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.12

Capitolo 6, Problema 5.RE.12

Use identities to write each expression in terms of sin θ and cos θ, and then simplify so that no quotients appear and all functions are of θ only.
csc² θ + sec² θ

Guida verificata passo dopo passo

Recall the definitions of the cosecant and secant functions in terms of sine and cosine: \(\csc \theta = \frac{1}{\sin \theta}\) and \(\sec \theta = \frac{1}{\cos \theta}\).

Rewrite the given expression \(\csc^{2} \theta + \sec^{2} \theta\) using these definitions: \(\left(\frac{1}{\sin \theta}\right)^{2} + \left(\frac{1}{\cos \theta}\right)^{2}\).

Simplify the expression to \(\frac{1}{\sin^{2} \theta} + \frac{1}{\cos^{2} \theta}\), which is a sum of two fractions.

Find a common denominator for the two fractions, which is \(\sin^{2} \theta \cos^{2} \theta\), and rewrite the expression as \(\frac{\cos^{2} \theta}{\sin^{2} \theta \cos^{2} \theta} + \frac{\sin^{2} \theta}{\sin^{2} \theta \cos^{2} \theta}\).

Combine the fractions into a single fraction: \(\frac{\cos^{2} \theta + \sin^{2} \theta}{\sin^{2} \theta \cos^{2} \theta}\), then use the Pythagorean identity \(\sin^{2} \theta + \cos^{2} \theta = 1\) to simplify the numerator.

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Reciprocal Trigonometric Identities

Reciprocal identities express functions like cosecant and secant in terms of sine and cosine: csc θ = 1/sin θ and sec θ = 1/cos θ. These allow rewriting expressions involving csc² θ and sec² θ into fractions with sin θ and cos θ in the denominator.

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Fundamental Trigonometric Identities

Pythagorean Identities

Pythagorean identities relate sine and cosine functions, such as sin² θ + cos² θ = 1. These identities help simplify expressions by replacing squared terms or combining fractions to eliminate quotients and reduce complexity.

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Pythagorean Identities

Algebraic Simplification of Trigonometric Expressions

After substituting reciprocal identities, algebraic techniques like finding common denominators and combining fractions are used to simplify the expression. The goal is to write the expression without quotients and only in terms of sin θ and cos θ.

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Simplifying Trig Expressions

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