Find the Exact Value tan((7pi)/4-pi/6)

Problem

tan((7*π)/4−π/6)

Solution

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

2. Assign the values A=(7*π)/4 and B=π/6

3. Evaluate the tangent of each individual angle.

tan((7*π)/4)=−1

tan(π/6)=√(,3)/3

4. Substitute these values into the difference formula.

tan((7*π)/4−π/6)=(−1−√(,3)/3)/(1+(−1)*(√(,3)/3))

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

tan((7*π)/4−π/6)=(−3−√(,3))/3/(3−√(,3))/3

6. Cancel the common denominator of 3

tan((7*π)/4−π/6)=(−3−√(,3))/(3−√(,3))

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

tan((7*π)/4−π/6)=((−3−√(,3))*(3+√(,3)))/((3−√(,3))*(3+√(,3)))

8. Expand the products in the numerator and denominator.

tan((7*π)/4−π/6)=(−9−3√(,3)−3√(,3)−3)/(9−3)

9. Combine like terms and simplify the fraction.

tan((7*π)/4−π/6)=(−12−6√(,3))/6

tan((7*π)/4−π/6)=−2−√(,3)

Final Answer

tan((7*π)/4−π/6)=−2−√(,3)

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