Use the given information to find cos(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 \(\cos y = \pm \sqrt{1 - \sin^2 y} = \pm \sqrt{1 - \left(-\frac{2}{3}\right)^2}\).

Determine the correct sign of \(\cos y\) based on the quadrant of \(y\). Since \(y\) is in quadrant III, both sine and cosine are negative, so \(\cos y\) is negative.

Similarly, find \(\sin x\) using the Pythagorean identity with \(\cos x = -\frac{1}{5}\). Calculate \(\sin x = \pm \sqrt{1 - \cos^2 x} = \pm \sqrt{1 - \left(-\frac{1}{5}\right)^2}\).

Determine the correct sign of \(\sin x\) based on the quadrant of \(x\). Since \(x\) is in quadrant II, sine is positive and cosine is negative, so \(\sin x\) is positive. Finally, use the cosine difference formula: \(\cos(x - y) = \cos x \cos y + \sin x \sin y\) to express \(\cos(x - y)\) in terms of the values found.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Trigonometric Identities for Cosine of a Difference

The cosine of the difference of two angles, cos(x - y), can be found using the identity cos(x - y) = cos x cos y + sin x sin y. This formula allows us to express cos(x - y) in terms of the sines and cosines of x and y individually.

Determining Signs of Trigonometric Functions by Quadrant

The signs of sine and cosine depend on the quadrant of the angle. 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 of sin x and cos y when only partial information is given.

Using the Pythagorean Identity to Find Missing Values

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

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