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.RE.32b

Capitolo 6, Problema 5.RE.32b

Use the given information to cos(x - y).
cos x = 2/9, sin y = -1/2, x in quadrant IV, y in quadrant III

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Identify the given information: \(\cos x = \frac{2}{9}\), \(\sin y = -\frac{1}{2}\), with \(x\) in quadrant IV and \(y\) in quadrant III.

Find \(\sin x\) using the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\). Since \(x\) is in quadrant IV, where sine is negative, calculate \(\sin x = -\sqrt{1 - \left(\frac{2}{9}\right)^2}\).

Find \(\cos y\) using the Pythagorean identity \(\sin^2 y + \cos^2 y = 1\). Since \(y\) is in quadrant III, where cosine is negative, calculate \(\cos y = -\sqrt{1 - \left(-\frac{1}{2}\right)^2}\).

Use the cosine difference formula: \(\cos(x - y) = \cos x \cos y + \sin x \sin y\).

Substitute the values of \(\cos x\), \(\cos y\), \(\sin x\), and \(\sin y\) into the formula and simplify the expression to find \(\cos(x - y)\).

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Trigonometric Identities - Cosine of a Difference

The cosine of a difference formula states that cos(x - y) = cos x cos y + sin x sin y. This identity allows us to express the cosine of the difference of two angles in terms of the sines and cosines of the individual angles, which is essential for solving the problem.

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Sum and Difference of Sine & Cosine

Determining Signs of Trigonometric Functions by Quadrant

The signs of sine and cosine depend on the quadrant of the angle. In quadrant IV, cosine is positive and sine is negative; in quadrant III, both sine and cosine are negative. Knowing the quadrant helps assign correct signs to sin x and cos y when they are not directly given.

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Introduction to Trigonometric Functions

Using Pythagorean Identity to Find Missing Values

The Pythagorean identity, sin²θ + cos²θ = 1, allows calculation of an unknown sine or cosine value when the other is known. For example, if cos x is given, sin x can be found by sin x = ±√(1 - cos²x), with the sign determined by the quadrant.

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Pythagorean Identities

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