Find the Exact Value tan((11pi)/12)

Problem

tan((11*π)/12)

Solution

1. Identify the angle as a sum or difference of known special angles. We can write (11*π)/12 as (3*π)/12+(8*π)/12

2. Simplify the fractions to standard angles in radians.

(3*π)/12=π/4

(8*π)/12=(2*π)/3

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

tan(π/4+(2*π)/3)=(tan(π/4)+tan((2*π)/3))/(1−tan(π/4)*tan((2*π)/3))

4. Substitute the known values tan(π/4)=1 and tan((2*π)/3)=−√(,3)

(1+(−√(,3)))/(1−(1)*(−√(,3)))

5. Simplify the expression.

(1−√(,3))/(1+√(,3))

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

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

7. Expand the numerator and denominator.

(1−2√(,3)+3)/(1−3)

8. Simplify the resulting fraction.

(4−2√(,3))/(−2)

−2+√(,3)

Final Answer

tan((11*π)/12)=√(,3)−2

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