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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.RE.30a
Capitolo 6, Problema 5.RE.30a

Use the given information to find sin(x + y).
sin y = - 2/3 , cos x = - 1/5, x in quadrant II, y in quadrant III

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Identify the given information: \(\sin y = -\frac{2}{3}\), \(\cos x = -\frac{1}{5}\), with \(x\) in quadrant II and \(y\) in quadrant III.
Use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) to find \(\cos y\). Since \(\sin y = -\frac{2}{3}\), calculate \(\cos y = \pm \sqrt{1 - \sin^2 y} = \pm \sqrt{1 - \left(-\frac{2}{3}\right)^2}\), and determine the correct sign based on the quadrant of \(y\).
Similarly, find \(\sin x\) using \(\cos x = -\frac{1}{5}\). Calculate \(\sin x = \pm \sqrt{1 - \cos^2 x} = \pm \sqrt{1 - \left(-\frac{1}{5}\right)^2}\), and choose the sign according to the quadrant of \(x\).
Recall the angle addition formula for sine: \(\sin(x + y) = \sin x \cos y + \cos x \sin y\).
Substitute the values of \(\sin x\), \(\cos y\), \(\cos x\), and \(\sin y\) into the formula to express \(\sin(x + y)\) in terms of known quantities.

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Sum of Angles Formula for Sine

The sine of the sum of two angles, sin(x + y), can be found using the formula sin(x + y) = sin x cos y + cos x sin y. This identity allows us to express sin(x + y) in terms of the sines and cosines of the individual angles x and y.
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Verifying Identities with Sum and Difference Formulas

Determining Signs of Trigonometric Functions by Quadrant

The signs of sine and cosine depend on the quadrant in which the angle lies. In quadrant II, sine is positive and cosine is negative; in quadrant III, both sine and cosine are negative. This helps determine the correct values and signs of unknown trigonometric functions.
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Introduction to Trigonometric Functions

Using Pythagorean Identity to Find Missing Values

The Pythagorean identity, sin²θ + cos²θ = 1, allows us to find an unknown sine or cosine value when the other is given. By substituting the known value and solving, we can find the missing trigonometric function needed to apply the sum formula.
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Pythagorean Identities
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