Final answer:
To calculate tan(x + y), we use the tangent sum identity. However, after a step-by-step calculation, the decimal answer 1.73 is obtained, which is not among the provided options, indicating a possible mistake in the question or the options.
Step-by-step explanation:
The student is asking about finding the value of the tangent of the sum of two angles, where the sine of one angle, x, is given as 1/2, and the cosine of another angle, y, is given as 2/√2. Since both angles are in the first quadrant, their trigonometric ratios are positive.
To find tan (x + y), we use the tangent sum identity, which is tan (x + y) = (tan x + tan y) / (1 - tan x * tan y). We have sin(x) = 1/2, so using the Pythagorean identity, we can find cos(x) = √(1 - sin2(x)) = √(1 - 1/4) = √(3/4) and tan(x) = sin(x) / cos(x) = (1/2) / (√3/2) = 1/√3.
Also, cos(y) = 2/√2, so sin(y) = √(1 - cos2(y)) = √(1 - 2/2) = √0, which is not possible since sin(y) must be positive in the first quadrant. We need to correct this to sin(y) = √(1 - (2/√2)2) = √(1 - 1) = 0. Thus, tan(y) = sin(y) / cos(y) = 0 / (2/√2) = 0.
Now, applying the tangent sum formula, tan (x + y) = (1/√3 + 0) / (1 - (1/√3 * 0)) = 1/√3. Converting to a decimal, tan (x + y) = 1.73, which is not present in the given options A through D, suggesting a possible error in the options provided or a misunderstanding in the initial conditions.