Evaluate tan(pi/12)

Problem

tan(π/12)

Solution

1. Identify the angle in terms of known values. The angle π/12 can be written as the difference between 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 trigonometric values tan(π/4)=1 and tan(π/6)=√(,3)/3

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. Simplify the resulting expression.

tan(π/12)=(12−6√(,3))/6

8. Divide each term by 6 to reach the final result.

tan(π/12)=2−√(,3)

Final Answer

tan(π/12)=2−√(,3)

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