Evaluate cot(pi/12)

Problem

cot(π/12)

Solution

1. Identify the angle in terms of known values. The angle π/12 can be expressed as the difference between two common angles: π/3−π/4 or π/4−π/6 We will use π/4−π/6

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

3. Substitute the values A=π/4 and B=π/6 into the formula.

cot(π/4−π/6)=(cot(π/4)*cot(π/6)+1)/(cot(π/6)−cot(π/4))

4. Evaluate the trigonometric functions at these specific angles. We know cot(π/4)=1 and cot(π/6)=√(,3)

cot(π/12)=((1)*(√(,3))+1)/(√(,3)−1)

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

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

6. Simplify the expression. The numerator becomes 3+2√(,3)+1 and the denominator becomes 3 - 1$.

cot(π/12)=(4+2√(,3))/2

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

cot(π/12)=2+√(,3)

Final Answer

cot(π/12)=2+√(,3)

Want more problems? Check here!