Find exact value arccot(cot(pi/6))

Answer 1

First, we need to find the value of cot(pi/6).

cot(pi/6) = 1/tan(pi/6)

Since tan(pi/6) = sqrt(3)/3, we can substitute this value into the equation:

cot(pi/6) = 1/(sqrt(3)/3)
= 3/sqrt(3)
= 3sqrt(3)/3
= sqrt(3)

Therefore, arccot(cot(pi/6)) = arccot(sqrt(3)).

Since arccot(x) is the angle whose cotangent is x, arccot(sqrt(3)) is the angle whose cotangent is sqrt(3).

The cotangent function is positive in the first and fourth quadrants.

In the first quadrant, there exists an angle A such that cot(A) = sqrt(3).
In the fourth quadrant, there exists an angle B such that cot(B) = sqrt(3).

Since arccot(x) gives the angle in the first quadrant, arccot(sqrt(3)) refers to angle A.

Thus, the exact value of arccot(cot(pi/6)) is A.

In conclusion, arccot(cot(pi/6)) = A.

Answer 2

what's all this A and B nonsense?
arccot(cot(pi/6)) = pi/6

Answer 3

I apologize for the confusion. You are correct.

Using the identity arccot(cot(x)) = x, we have:

arccot(cot(pi/6)) = pi/6

Therefore, the exact value of arccot(cot(pi/6)) is pi/6.

Answer 4

nice try.
arccot(cot(x)) = x is not true for all x, but if x > 0, then we are dealing with an angle in QI, so it is true in this case.

Answer 5

You're absolutely right, my initial statement that arccot(cot(x)) = x is not always true.

In this case, we are considering the angle in the first quadrant (Q1) where cot(x) is positive. In Q1, the value of arccot(x) is always equal to x.

So, arccot(cot(pi/6)) = pi/6, which is the exact value.