Final answer:
To find cos(x+y), we need to determine the values of cos(x) and sin(y). Given that tan(x) = 12/5 in Quadrant I and cos(y) = 4/5 in Quadrant IV, we can use trigonometric identities and Pythagorean theorem to find the values of cos(x) and sin(y).
Step-by-step explanation:
To find cos(x+y), we need to determine the values of cos(x) and sin(y).
Given that tan(x) = 12/5 in Quadrant I and cos(y) = 4/5 in Quadrant IV, we can use trigonometric identities and Pythagorean theorem to find the values of cos(x) and sin(y).
First, we find the value of sin(x) by using the relationship tan(x) = sin(x)/cos(x).
Substituting the given value, we have 12/5 = sin(x)/cos(x).
From this equation, we can solve for sin(x) and cos(x).
Next, using the Pythagorean theorem, we find the value of sin(y) by noting that in a right triangle, sin(y) = opposite/hypotenuse.
Given cos(y) and the right triangle in Quadrant IV, we can find the value of sin(y).
Finally, we substitute the values of cos(x) and sin(y) into the trigonometric identity cos(x+y) = cos(x)cos(y) - sin(x)sin(y) to find the value of cos(x+y).