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.