Final answer:
To find cos(x) when sin(x) = 2/3 and sec(x) < 0, we can use the Pythagorean identity sin^2(x) + cos^2(x) = 1. Using this identity, we solve for cos(x) and find that cos(x) = -sqrt(5/9).
Step-by-step explanation:
To find the value of cos(x) when sin(x) = 2/3 and sec(x) < 0, we can use the Pythagorean identity sin^2(x) + cos^2(x) = 1. We know that sin(x) = 2/3, so sin^2(x) = (2/3)^2 = 4/9. Plugging this into the Pythagorean identity, we get cos^2(x) = 1 - 4/9 = 5/9. Taking the square root of both sides, we find that cos(x) = +/- sqrt(5/9). Since sec(x) is negative, cos(x) must also be negative. Therefore, cos(x) = -sqrt(5/9).