Simplify tan(pi/12)

Problem

tan(π/12)

Solution

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

π/12=π/3−π/4

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/3−π/4)=(tan(π/3)−tan(π/4))/(1+tan(π/3)*tan(π/4))

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

(√(,3)−1)/(1+√(,3)*(1))

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

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

5. Expand the numerator and denominator.

(√(,3)−3−1+√(,3))/(1−3)

6. Simplify the resulting expression.

(2√(,3)−4)/(−2)

7. Divide each term by −2 to find the final value.

2−√(,3)

Final Answer

tan(π/12)=2−√(,3)

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