Find the Exact Value tan(5/12)

Problem

tan((5*π)/12)

Solution

1. Identify the angle in degrees to make it easier to recognize common values.

(5*π)/12⋅180/π=75

2. Rewrite the angle as a sum of two special angles whose trigonometric values are known.

75=45+30

(5*π)/12=π/4+π/6

3. Apply the formula for the tangent of a sum, 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))

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

tan((5*π)/12)=(1+√(,3)/3)/(1−(1)*(√(,3)/3))

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

tan((5*π)/12)=(3+√(,3))/(3−√(,3))

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

tan((5*π)/12)=((3+√(,3))2)/((3−√(,3))*(3+√(,3)))

tan((5*π)/12)=(9+6√(,3)+3)/(9−3)

tan((5*π)/12)=(12+6√(,3))/6

tan((5*π)/12)=2+√(,3)

Final Answer

tan((5*π)/12)=2+√(,3)

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