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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