Expand Using Sum/Difference Formulas tan(pi/12)

Problem

tan(π/12)

Solution

1. Rewrite the angle as a difference of two common angles from the unit circle.

π/12=(4*π)/12−(3*π)/12

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))

(√(,3)−1)/(1+√(,3))

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

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

5. Simplify the expression by expanding the terms.

(√(,3)−3−1+√(,3))/(1−3)

(2√(,3)−4)/(−2)

2−√(,3)

Final Answer

tan(π/12)=2−√(,3)

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