Find the Exact Value cot(-(5pi)/12)

Problem

cot(−(5*π)/12)

Solution

1. Use the odd-even property of the cotangent function, which states cot(−θ)=−cot(θ)

cot(−(5*π)/12)=−cot((5*π)/12)

2. Rewrite the angle (5*π)/12 as a sum of two special angles from the unit circle.

(5*π)/12=(3*π)/12+(2*π)/12

(5*π)/12=π/4+π/6

3. Apply the cotangent sum formula, which is cot(A+B)=(cot(A)*cot(B)−1)/(cot(B)+cot(A))

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

4. Substitute the known values cot(π/4)=1 and cot(π/6)=√(,3)

−((1)*(√(,3))−1)/(√(,3)+1)=−(√(,3)−1)/(√(,3)+1)

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

−((√(,3)−1)*(√(,3)−1))/((√(,3)+1)*(√(,3)−1))

−(3−2√(,3)+1)/(3−1)

−(4−2√(,3))/2

6. Simplify the expression by dividing each term in the numerator by 2.

−(2−√(,3))

√(,3)−2

Final Answer

cot(−(5*π)/12)=√(,3)−2

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