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

Problem

tan((5*π)/12)

Solution

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

(5*π)/12=(3*π)/12+(2*π)/12

(5*π)/12=π/4+π/6

2. Apply the sum formula for tangent, which is tan(A+B)=(tan(A)+tan(B))/(1−tan(A)*tan(B))

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

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

(1+√(,3)/3)/(1−(1)*(√(,3)/3))

4. Simplify the complex fraction by multiplying the numerator and denominator by 3

(3+√(,3))/(3−√(,3))

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

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

(9+6√(,3)+3)/(9−3)

(12+6√(,3))/6

2+√(,3)

Final Answer

tan((5*π)/12)=2+√(,3)

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