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 5.22

Capitolo 6, Problema 5.22

Perform each indicated operation and simplify the result so that there are no quotients.
1/( sin α - 1) - 1/(sin α + 1)

Guida verificata passo dopo passo

Identify a common denominator for the two fractions. The common denominator is \((\sin \alpha - 1)(\sin \alpha + 1)\).

Rewrite each fraction with the common denominator: \(\frac{1}{\sin \alpha - 1} = \frac{\sin \alpha + 1}{(\sin \alpha - 1)(\sin \alpha + 1)}\) and \(\frac{1}{\sin \alpha + 1} = \frac{\sin \alpha - 1}{(\sin \alpha - 1)(\sin \alpha + 1)}\).

Subtract the two fractions: \(\frac{\sin \alpha + 1}{(\sin \alpha - 1)(\sin \alpha + 1)} - \frac{\sin \alpha - 1}{(\sin \alpha - 1)(\sin \alpha + 1)}\).

Combine the numerators over the common denominator: \(\frac{(\sin \alpha + 1) - (\sin \alpha - 1)}{(\sin \alpha - 1)(\sin \alpha + 1)}\).

Simplify the numerator: \((\sin \alpha + 1) - (\sin \alpha - 1) = 2\), resulting in \(\frac{2}{(\sin \alpha - 1)(\sin \alpha + 1)}\).

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Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate angles to the ratios of sides in right triangles. In this question, the sine function is used, which is defined as the ratio of the length of the opposite side to the hypotenuse in a right triangle. Understanding how these functions behave and their properties is essential for manipulating expressions involving them.

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Percorso guidato

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Introduction to Trigonometric Functions

Common Denominator

When performing operations with fractions, finding a common denominator is crucial for combining them. In this case, the two fractions have different denominators: (sin α - 1) and (sin α + 1). To simplify the expression, one must find a common denominator, which is the product of the two denominators, allowing for the combination of the fractions into a single expression.

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Percorso guidato

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Rationalizing Denominators

Simplification of Expressions

Simplification involves reducing an expression to its simplest form, often by eliminating common factors or combining like terms. In this problem, after finding a common denominator and combining the fractions, further simplification may involve factoring or canceling terms. This process is essential to ensure the final result is presented without quotients, as specified in the question.

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Percorso guidato

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Simplifying Trig Expressions

Pratica correlata

Domanda del libro di testo

Advanced methods of trigonometry can be used to find the following exact value.

sin 18° = (√5 - 1)/4

(See Hobson's A Treatise on Plane Trigonometry.) Use this value and identities to find each exact value. Support answers with calculator approximations if desired.

csc 18°

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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.

csc² t - 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.

tan(-θ)/sec θ

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Find the exact value of each expression. (Do not use a calculator.)

cos(-15°)

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Verify that each equation is an identity.

(sec α - tan α)² = (1 - sin α)/(1 + sin α)

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