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

Problem

tan((11*π)/6−π/4)

Solution

1. Identify the appropriate trigonometric identity for the tangent of a difference, 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 to the formula.

3. Evaluate the individual tangent values using the unit circle.

tan((11*π)/6)=−√(,3)/3

tan(π/4)=1

4. Substitute these values into the difference formula.

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

5. Simplify the numerator and the denominator by finding a common denominator of 3

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

6. Cancel the common denominator of 3 and simplify the resulting fraction.

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)*(3+√(,3)))/((3−√(,3))*(3+√(,3)))

8. Expand the products in the numerator and denominator.

tan((11*π)/6−π/4)=(−3√(,3)−3−9−3√(,3))/(9−3)

9. Combine like terms and simplify the fraction.

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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