Find the Exact Value arccot(- square root of 3)

Problem

arccot(−√(,3))

Solution

1. Identify the definition of the inverse cotangent function. The value y=arccot(x) is the angle y in the interval (0,π) such that cot(y)=x

2. Set up the equation based on the given value. We need to find y such that:

cot(y)=−√(,3)

3. Relate cotangent to sine and cosine to find the reference angle. Since cot(y)=cos(y)/sin(y) we look for an angle where the ratio of cosine to sine is −√(,3)

cos(y)/sin(y)=−√(,3)

4. Determine the reference angle in the first quadrant. We know that cot(π/6)=√(,3) because cos(π/6)=√(,3)/2 and sin(π/6)=1/2

cot(π/6)=√(,3)

5. Apply the range of the arccotangent function. Since the input −√(,3) is negative, the angle y must be in the second quadrant, specifically (π/2,π)

y=π−π/6

6. Calculate the final value by subtracting the fractions.

y=(5*π)/6

Final Answer

arccot(−√(,3))=(5*π)/6

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