Find the Exact Value tan((-5pi)/12)

Problem

tan(−(5*π)/12)

Solution

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

tan(−(5*π)/12)=−tan((5*π)/12)

2. Rewrite the angle as a sum of two special angles whose trigonometric values are known.

(5*π)/12=(2*π)/12+(3*π)/12

(5*π)/12=π/6+π/4

3. Apply the tangent sum formula, tan(A+B)=(tan(A)+tan(B))/(1−tan(A)*tan(B)) where A=π/6 and B=π/4

tan(π/6+π/4)=(tan(π/6)+tan(π/4))/(1−tan(π/6)*tan(π/4))

4. Substitute the known values tan(π/6)=√(,3)/3 and tan(π/4)=1

tan((5*π)/12)=(√(,3)/3+1)/(1−√(,3)/3*(1))

5. Simplify the fraction by multiplying the numerator and denominator by 3

tan((5*π)/12)=(√(,3)+3)/(3−√(,3))

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

tan((5*π)/12)=((√(,3)+3)*(3+√(,3)))/((3−√(,3))*(3+√(,3)))

tan((5*π)/12)=(3√(,3)+3+9+3√(,3))/(9−3)

tan((5*π)/12)=(12+6√(,3))/6

tan((5*π)/12)=2+√(,3)

7. Apply the negative sign from the first step to find the final value.

tan(−(5*π)/12)=−(2+√(,3))

Final Answer

tan(−(5*π)/12)=−2−√(,3)

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