Use the given information to find tan(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: \(\cos x = \frac{2}{9}\) with \(x\) in quadrant IV, and \(\sin y = -\frac{1}{2}\) with \(y\) 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 angle addition formula for tangent: \(\tan(x + y) = \frac{\tan x + \tan y}{1 - \tan x \tan y}\). Calculate \(\tan x = \frac{\sin x}{\cos x}\) and \(\tan y = \frac{\sin y}{\cos y}\) using the values found.

Substitute \(\tan x\) and \(\tan y\) into the formula and simplify the expression to find \(\tan(x + y)\).

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Key Concepts

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

Trigonometric Ratios and Quadrants

Trigonometric ratios (sine, cosine, tangent) relate the angles of a triangle to side lengths. The sign of these ratios depends on the quadrant where the angle lies. For example, in quadrant IV, cosine is positive and sine is negative; in quadrant III, both sine and cosine are negative. Understanding these sign conventions is essential for correctly determining values of trigonometric functions.

Sum of Angles Formula for Tangent

The tangent of a sum of two angles, tan(x + y), can be found using the formula tan(x + y) = (tan x + tan y) / (1 - tan x * tan y). This formula allows the calculation of the tangent of a combined angle from the tangents of individual angles, which can be derived from sine and cosine values.

Finding Missing Trigonometric Values

Given partial trigonometric information (like cos x or sin y), the other values (sin x, cos y, tan x, tan y) can be found using the Pythagorean identity sin²θ + cos²θ = 1. The quadrant information helps determine the correct sign of these values. This step is crucial before applying the sum formula for tangent.

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