Final answer:
Using the trigonometric identities, we find that for sin x = -4/5 and cos x > 0, tan x is -4/3 and sec x is 5/3.
Step-by-step explanation:
To find tan x and sec x, we use the given information sin x = -4/5 and cos x > 0. With sin x known, we can find cos x by understanding that sin² x + cos² x = 1; this is one of the fundamental trigonometric identities. As cos x is positive and we know sin x, we can say cos x = √(1 - sin² x) = √(1 - (-4/5)²) = √(1 - 16/25) = √(9/25) = 3/5.
Since both sine and cosine are now known, tan x = sin x / cos x which equals (-4/5) / (3/5) = -4/3. Similarly, sec x = 1 / cos x equals 1 / (3/5) = 5/3.
Therefore, tan x is -4/3 and sec x is 5/3 given the conditions that sin x = -4/5 and cos x > 0.