Find the Exact Value tan((5pi)/3-pi/4)

Problem

tan((5*π)/3−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=(5*π)/3 and B=π/4 to the identity.

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

tan((5*π)/3)=−√(,3)

tan(π/4)=1

4. Substitute these values into the difference formula.

tan((5*π)/3−π/4)=(−√(,3)−1)/(1+(−√(,3))*(1))

5. Simplify the expression by organizing the terms in the numerator and denominator.

tan((5*π)/3−π/4)=(−√(,3)−1)/(1−√(,3))

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

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

7. Expand the numerator and denominator.

Numerator: *(−√(,3)−1)*(1+√(,3))=−√(,3)−3−1−√(,3)=−4−2√(,3)

Denominator: *(1−√(,3))*(1+√(,3))=1−3=−2

8. Divide the terms to find the final simplified value.

(−4−2√(,3))/(−2)=2+√(,3)

Final Answer

tan((5*π)/3−π/4)=2+√(,3)

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