Find the Exact Value cot(pi/12)

Problem

cot(π/12)

Solution

1. Identify the angle in terms of known values. The angle π/12 can be written as the difference between π/3 and π/4

π/12=(4*π−3*π)/12=π/3−π/4

2. Apply the identity for the cotangent of a difference. The formula is cot(A−B)=(cot(A)*cot(B)+1)/(cot(B)−cot(A))

cot(π/3−π/4)=(cot(π/3)*cot(π/4)+1)/(cot(π/4)−cot(π/3))

3. Substitute the known trigonometric values. We know cot(π/3)=1/√(,3) and cot(π/4)=1

cot(π/12)=(1/√(,3)*(1)+1)/(1−1/√(,3))

4. Simplify the complex fraction by multiplying the numerator and denominator by √(,3)

cot(π/12)=(1+√(,3))/(√(,3)−1)

5. Rationalize the denominator by multiplying the numerator and denominator by the conjugate √(,3)+1

cot(π/12)=((1+√(,3))*(√(,3)+1))/((√(,3)−1)*(√(,3)+1))

6. Expand the numerator and denominator.

cot(π/12)=(3+2√(,3)+1)/(3−1)

cot(π/12)=(4+2√(,3))/2

7. Divide each term by 2 to reach the final result.

cot(π/12)=2+√(,3)

Final Answer

cot(π/12)=2+√(,3)

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