If sin theta = -7/8, then the value of 1/cot theta in the interval 3pi/2 <theta <2pi is:

To find the value of 1/cot(theta) in the interval 3pi/2 < theta < 2pi, we need to determine the value of cot(theta) first.

We know that cot(theta) is the reciprocal of tan(theta). We can use the formula: cot(theta) = 1/tan(theta).

Given that sin(theta) = -7/8, we can determine cos(theta) using the Pythagorean identity: sin^2(theta) + cos^2(theta) = 1.

sin^2(theta) = (-7/8)^2 = 49/64.

Now, substitute sin^2(theta) into the Pythagorean identity:

49/64 + cos^2(theta) = 1.

Move cos^2(theta) to the other side:

cos^2(theta) = 1 - 49/64.

cos^2(theta) = 64/64 - 49/64.

cos^2(theta) = 15/64.

Taking the square root of both sides and since we're in the interval 3pi/2 < theta < 2pi, we can determine that cos(theta) = -√(15/64) or cos(theta) = √(15/64).

Now, to find cot(theta), we use the formula cot(theta) = 1/tan(theta).

tan(theta) = sin(theta) / cos(theta).

For sin(theta) = -7/8 and cos(theta) = √(15/64), we have:

tan(theta) = (-7/8) / (√(15/64)).

tan(theta) = (-7/8) / (√15/8).

Dividing the numerator and denominator by 8:

tan(theta) = -7 / (√15).

Finally, to find 1/cot(theta), we take the reciprocal of cot(theta):

1 / cot(theta) = 1 / (-7 / (√15)).

Multiplying the numerator and denominator by (√15):

1 / cot(theta) = (√15) / (-7).

Therefore, the value of 1/cot(theta) in the interval 3pi/2 < theta < 2pi is -(√15)/7.