Find the Exact Value tan((-7pi)/12)

Problem

tan(−(7*π)/12)

Solution

1. Use the odd function property of the tangent function, which states that tan(−θ)=−tan(θ)

−tan((7*π)/12)

2. Rewrite the angle as a sum of two standard angles from the unit circle.

(7*π)/12=(3*π)/12+(4*π)/12=π/4+π/3

3. Apply the sum formula for tangent, tan(A+B)=(tan(A)+tan(B))/(1−tan(A)*tan(B))

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

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

−((1+√(,3))/(1−(1)*(√(,3))))

5. Simplify the expression by distributing the negative sign into the denominator.

(1+√(,3))/(√(,3)−1)

6. Rationalize the denominator by multiplying the numerator and denominator by the conjugate √(,3)+1

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

7. Expand and simplify the numerator and denominator.

(3+√(,3)+√(,3)+1)/(3−1)

(4+2√(,3))/2

2+√(,3)

Final Answer

tan(−(7*π)/12)=2+√(,3)

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