Final answer:
The value of cos(s+t) is (-sqrt(3)-sqrt(2))/6.
Step-by-step explanation:
To find cos(s+t), we need to use the trigonometric identity for cosine of the sum of two angles:
cos(s+t) = cos(s)cos(t) - sin(s)sin(t)
Given that cos(s) = 1/3 and sin(t) = -1/2, we can substitute these values into the formula:
cos(s+t) = (1/3)(cos(t)) - (sin(s))(1/2) = (1/3)(cos(t)) - (1/2)(sqrt(1 - cos^2(s)))
Since s is in quadrant 1 and cos(s) = 1/3, we can determine that sin(s) = sqrt(1 - cos^2(s)) = sqrt(1 - (1/3)^2) = sqrt(8/9) = sqrt(2/3).
Now, we need to determine cos(t) which requires knowing the quadrant of t. Given that t is in quadrant 3 and sin(t) = -1/2, we can determine that cos(t) = -sqrt(1 - sin^2(t)) = -sqrt(1 - (-1/2)^2) = -sqrt(3/4) = -sqrt(3)/2.
Substituting the values of sin(s), cos(t) into the formula, we have:
cos(s+t) = (1/3)(-sqrt(3)/2) - (1/2)(sqrt(2/3)) = -sqrt(3)/6 - sqrt(2)/6 = (-sqrt(3)-sqrt(2))/6.