Find the Exact Value tan((7pi)/4-pi/3)

Problem

tan((7*π)/4−π/3)

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=(7*π)/4 and B=π/3 to the formula.

3. Evaluate the individual tangent values using the unit circle: tan((7*π)/4)=−1 and tan(π/3)=√(,3)

4. Substitute these values into the identity:

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

5. Simplify the expression:

tan((7*π)/4−π/3)=(−1−√(,3))/(1−√(,3))

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

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

7. Expand the numerator and denominator:

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

8. Combine like terms:

tan((7*π)/4−π/3)=(−4−2√(,3))/(−2)

9. Divide each term in the numerator by −2

tan((7*π)/4−π/3)=2+√(,3)

Final Answer

tan((7*π)/4−π/3)=2+√(,3)

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