Find all values of x, 0<x<(ƒÎ/2) for which (1/�ã3)<cotx<(�ã3)
To find all values of x, 0 < x < π/2 (ƒÎ/2) for which (1/√3)<cotx<(√3), we can use the properties of the cotangent function.
First, let's rewrite the given inequality using the reciprocal identity for cotangent:
(1/√3) < cot(x) < √3
Now, let's take the inverse cotangent (arccot) of all three parts of the inequality:
arccot(1/√3) < x < arccot(√3)
Now we just need to find the values of arccot(1/√3) and arccot(√3) within the given range 0 < x < π/2.
Using a calculator, we can evaluate these inverse cotangent values:
arccot(1/√3) ≈ 0.5236 radians (≈ 30 degrees)
arccot(√3) ≈ 0.7854 radians (≈ 45 degrees)
So, the solution for 0 < x < π/2 is:
0.5236 radians < x < 0.7854 radians
or
30 degrees < x < 45 degrees.