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

Problem

cot((5*π)/12)

Solution

1. Identify the angle as a sum of two known special angles from the unit circle.

(5*π)/12=(2*π)/12+(3*π)/12

2. Simplify the fractions to express the angle in terms of standard values.

(5*π)/12=π/6+π/4

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

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

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

((√(,3))*(1)−1)/(1+√(,3))

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

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

6. Expand the numerator and denominator.

(3−√(,3)−√(,3)+1)/(3−1)

7. Simplify the resulting fraction.

(4−2√(,3))/2

8. Divide each term in the numerator by the denominator.

2−√(,3)

Final Answer

cot((5*π)/12)=2−√(,3)

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