Find the Exact Value tan(pi/12)

Problem

tan(π/12)

Solution

1. Identify a way to rewrite the angle π/12 using common angles from the unit circle, such as π/4 and π/6

π/12=π/4−π/6

2. Apply the formula for the tangent of a difference, 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 into the expression.

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

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

tan(π/12)=(3−√(,3))/(3+√(,3))

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

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

6. Expand the numerator and denominator.

tan(π/12)=(9−6√(,3)+3)/(9−3)

7. Combine like terms and simplify the resulting fraction.

tan(π/12)=(12−6√(,3))/6

tan(π/12)=2−√(,3)

Final Answer

tan(π/12)=2−√(,3)

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