Remark
The cot(theta) = 4/7
That means we have an adjacent of 4 and an opposite of 7. We still need to find the hypotenuse.
Step One
Find the hypotenuse
a^2 + b^2 = c^2
a = 4
b = 7
c = ???
4^2 + 7^2 = c^2
c^2 = 16 + 49
c^2 = 65 Take the square root of both sides.
c = sqrt(65)
Now find Sin(theta)
Sin(theta) = 7/ sqrt(65)
The other way to do it would be to solve
cot^2(theta) + 1 = csc^2(theta)
(4/7)^2 + 1 = csc^2(theta)
16/49 + 1 = csc^2 (theta)
65/49 =csc^2(theta)
sqrt(65)/7 = csc(theta)
7 / sqrt(65) = sin(theta)
There's a choice of two ways to do it. The first way is using cot = adjacent /opposite.
The second is using
1 + cot^2(theta) = csc^2(theta) and
sin(theta) = 1 / csc(theta).