Ch. 5 - Trigonometric Identities

Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo

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Lial 12th Edition

Ch. 5 - Trigonometric Identities

Problema 5.1.22

Capitolo 6, Problema 5.1.22

Find sinθ.
csc θ = -8/5

Guida verificata passo dopo passo

Recall the definition of cosecant: \(\csc \theta = \frac{1}{\sin \theta}\). This means \(\sin \theta = \frac{1}{\csc \theta}\).

Substitute the given value of \(\csc \theta = -\frac{8}{5}\) into the equation: \(\sin \theta = \frac{1}{-\frac{8}{5}}\).

Simplify the fraction by multiplying numerator and denominator: \(\sin \theta = -\frac{5}{8}\).

Consider the sign of \(\sin \theta\) based on the quadrant where \(\theta\) lies. Since \(\csc \theta\) is negative, \(\sin \theta\) is also negative, which matches the simplified value.

Conclude that \(\sin \theta = -\frac{5}{8}\), consistent with the given \(\csc \theta\).

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Reciprocal Trigonometric Functions

The cosecant function (csc θ) is the reciprocal of the sine function (sin θ), meaning csc θ = 1/sin θ. To find sin θ when given csc θ, you take the reciprocal of the given value.

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Percorso guidato

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Introduction to Trigonometric Functions

Sign of Trigonometric Functions in Quadrants

The sign of sine and cosecant depends on the quadrant where the angle θ lies. Since csc θ = -8/5 is negative, sin θ must also be negative, indicating θ is in either the third or fourth quadrant.

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Quadratic Formula

Simplifying and Interpreting Fractional Values

Given csc θ = -8/5, sin θ is the reciprocal, so sin θ = -5/8. Understanding how to simplify and interpret these fractional values is essential for accurate calculation and further problem-solving.

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4:02

Solving Linear Equations with Fractions

Pratica correlata

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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.

(cot² t - 1)/(1 + cot² t) = 1 - 2 sin² t

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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.

csc θ - sin θ

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Find the remaining five trigonometric functions of θ.

csc θ = -5/2, θ in quadrant III

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Verify that each equation is an identity.

tan α/sec α = sin α

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