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Ch. 1 - Trigonometric Functions
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. 1 - Trigonometric FunctionsProblema 23
Capitolo 2, Problema 23

Use the appropriate reciprocal identity to find each function value. Rationalize denominators when applicable. See Example 1.
sin θ , given that csc θ = 1.25

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1
Recall the reciprocal identity relating sine and cosecant: \(\csc \theta = \frac{1}{\sin \theta}\).
Given \(\csc \theta = 1.25\), rewrite this as \(1.25 = \frac{1}{\sin \theta}\).
To find \(\sin \theta\), take the reciprocal of both sides: \(\sin \theta = \frac{1}{1.25}\).
Simplify the fraction \(\frac{1}{1.25}\) by expressing 1.25 as a fraction: \(1.25 = \frac{5}{4}\), so \(\sin \theta = \frac{1}{\frac{5}{4}}\).
Use the property of dividing by a fraction: \(\sin \theta = 1 \times \frac{4}{5} = \frac{4}{5}\). This fraction is already rationalized.

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Reciprocal Identities

Reciprocal identities relate trigonometric functions to their reciprocals, such as sine and cosecant. Specifically, sin θ = 1 / csc θ. Understanding this allows you to find one function value when given its reciprocal.
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Pythagorean Identities

Rationalizing Denominators

Rationalizing denominators involves eliminating any irrational numbers from the denominator of a fraction. This is done by multiplying numerator and denominator by a suitable expression, making the expression simpler and more standard.
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Rationalizing Denominators

Evaluating Trigonometric Functions from Given Values

When given a trigonometric function value, such as csc θ, you can find related functions like sin θ by applying identities and performing algebraic manipulations. This process often requires careful substitution and simplification.
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Evaluate Composite Functions - Values Not on Unit Circle
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