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

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

Use the Pythagorean identity to find \(\sin x\). Since \(\sin^2 x + \cos^2 x = 1\), calculate \(\sin x = \pm \sqrt{1 - \cos^2 x} = \pm \sqrt{1 - \left(\frac{2}{9}\right)^2}\).

Determine the correct sign of \(\sin x\) based on the quadrant of \(x\). In quadrant IV, sine is negative, so \(\sin x\) is negative.

Find \(\cos y\) using the Pythagorean identity \(\sin^2 y + \cos^2 y = 1\). Since \(\sin y = -1\), calculate \(\cos y = \pm \sqrt{1 - (-1)^2} = 0\). Determine the sign of \(\cos y\) based on quadrant III, where cosine is negative, so \(\cos y = 0\).

Use the sine addition formula \(\sin(x + y) = \sin x \cos y + \cos x \sin y\) and substitute the values found for \(\sin x\), \(\cos y\), \(\cos x\), and \(\sin y\) to express \(\sin(x + y)\).

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Key Concepts

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

Trigonometric Angle Sum Identity

The sine of the sum of two angles, sin(x + y), can be found using the identity sin(x + y) = sin x cos y + cos x sin y. This formula allows us to express the sine of a combined angle in terms of the sines and cosines of the individual angles.

Determining Sine and Cosine Values from Quadrants

The signs of sine and cosine depend on the quadrant of the angle. In quadrant IV, sine is negative and cosine is positive; in quadrant III, both sine and cosine are negative. This helps determine the correct signs of unknown trigonometric values when given partial information.

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 rearranging the identity and considering the quadrant to assign the correct sign.

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