given cot-A square root of 5, find cot-B at the 4th quadrant.
To find cotangent (cot) of B in the fourth quadrant, we need to use the fact that cot is an even function, which means that it has the same value in the first and third quadrants. Since cot(A) = √5, we can determine the value of cot(B) by finding the cotangent of the reference angle of B in the first quadrant.
First, we need to find the reference angle of B. The reference angle is the acute angle formed between the terminal side of the angle and the x-axis in the standard position. In the fourth quadrant, the reference angle is the angle formed between the x-axis and a line that is parallel to the terminal side of the given angle.
In this case, since cot(A) = √5, we know that cos(A) = 1/sqrt(1 + cot^2(A)) = 1/sqrt(1 + 5) = 1/sqrt(6).
Since the cosine of an angle is positive in the fourth quadrant, we know that cos(B) = cos(pi - B) = 1/sqrt(6).
Next, we can use the reciprocal identity to find cot(B). The reciprocal identity for cotangent is cot(B) = 1/tan(B), and since tan(B) = sin(B)/cos(B), we can substitute the values we found earlier to get:
cot(B) = 1/(sin(B)/cos(B)) = cos(B)/sin(B) = (1/sqrt(6))/sin(B).
To find sin(B), we can use the Pythagorean identity for sine and cosine. Since cos(B) = 1/sqrt(6), we have sin(B) = sqrt(1 - cos^2(B)) = sqrt(1 - 1/6) = sqrt(5/6).
Now we have all the values we need to compute cot(B):
cot(B) = (1/sqrt(6))/sin(B) = (1/sqrt(6))/(sqrt(5/6)) = (1/sqrt(6)) * (sqrt(6/5)) = 1/√(5/6) = √(6/5) / 5.
Therefore, cot(B) in the fourth quadrant is √(6/5) / 5.