Find the Exact Value tan((11pi)/6+pi/4)

Problem

tan((11*π)/6+π/4)

Solution

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

2. Assign the values A=(11*π)/6 and B=π/4

3. Evaluate the tangent of each individual angle using the unit circle.

tan((11*π)/6)=−√(,3)/3

tan(π/4)=1

4. Substitute these values into the sum formula.

tan((11*π)/6+π/4)=(−√(,3)/3+1)/(1−(−√(,3)/3)*(1))

5. Simplify the numerator and the denominator.

tan((11*π)/6+π/4)=(3−√(,3))/3/(3+√(,3))/3

6. Cancel the common denominator of 3.

tan((11*π)/6+π/4)=(3−√(,3))/(3+√(,3))

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

tan((11*π)/6+π/4)=((3−√(,3))2)/(3−(√(,3))2)

8. Expand and simplify the resulting expression.

tan((11*π)/6+π/4)=(9−6√(,3)+3)/(9−3)

tan((11*π)/6+π/4)=(12−6√(,3))/6

tan((11*π)/6+π/4)=2−√(,3)

Final Answer

tan((11*π)/6+π/4)=2−√(,3)

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