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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 5.72
Capitolo 6, Problema 5.72

Verify that each equation is an identity.
tan² α sin² α = tan² α + cos² α - 1

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1
Start by expressing \( \tan^2 \alpha \) in terms of \( \sin \alpha \) and \( \cos \alpha \). Recall that \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \), so \( \tan^2 \alpha = \frac{\sin^2 \alpha}{\cos^2 \alpha} \).
Substitute \( \tan^2 \alpha = \frac{\sin^2 \alpha}{\cos^2 \alpha} \) into the left side of the equation: \( \frac{\sin^2 \alpha}{\cos^2 \alpha} \sin^2 \alpha \).
Simplify the left side: \( \frac{\sin^4 \alpha}{\cos^2 \alpha} \).
Now, consider the right side of the equation: \( \tan^2 \alpha + \cos^2 \alpha - 1 \). Substitute \( \tan^2 \alpha = \frac{\sin^2 \alpha}{\cos^2 \alpha} \) into the right side.
Simplify the right side: \( \frac{\sin^2 \alpha}{\cos^2 \alpha} + \cos^2 \alpha - 1 \). Use the Pythagorean identity \( \sin^2 \alpha + \cos^2 \alpha = 1 \) to simplify further and verify if both sides are equal.

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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. Understanding these identities is crucial for verifying equations and simplifying trigonometric expressions.
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Fundamental Trigonometric Identities

Tangent and Sine Functions

The tangent function, defined as the ratio of sine to cosine (tan α = sin α / cos α), plays a key role in trigonometry. The sine function represents the ratio of the opposite side to the hypotenuse in a right triangle. Recognizing how these functions relate to each other is essential for manipulating and verifying trigonometric equations.
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Sine, Cosine, & Tangent of 30°, 45°, & 60°

Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (a² + b² = c²). This theorem underpins many trigonometric identities, particularly the relationship between sine, cosine, and tangent. It is fundamental for deriving and verifying identities involving these functions.
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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.

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Verify that each equation is an identity.

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Factor each trigonometric expression.

sin³ α + cos³ α

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Simplify each expression.

±√[(1 - cos 8θ)/(1 + cos 8θ)]

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Perform each indicated operation and simplify the result so that there are no quotients.

(1 + tan θ)² - 2 tan θ

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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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