Answer:
To find cot(theta), we can use the trigonometric identity: cot(theta) = cos(theta) / sin(theta).
Given that sin(theta) = 2/3 and cos(theta) > 0, we can substitute these values into the formula:
cot(theta) = cos(theta) / sin(theta)
Since cos(theta) > 0, it means that cosine is positive, and we can determine the value of cos(theta) using the Pythagorean identity:
cos(theta) = sqrt(1 - sin^2(theta))
Let's substitute the value of sin(theta) into this equation:
cos(theta) = sqrt(1 - (2/3)^2)
cos(theta) = sqrt(1 - 4/9)
cos(theta) = sqrt(5/9)
cos(theta) = sqrt(5) / 3
Now we can substitute the values of cos(theta) and sin(theta) into the formula for cot(theta):
cot(theta) = (sqrt(5) / 3) / (2/3)
Simplifying:
cot(theta) = (sqrt(5) / 3) * (3/2)
cot(theta) = sqrt(5) / 2
Therefore, cot(theta) = sqrt(5) / 2.
Hope that helps!