Find the Exact Value tan((17pi)/12)

Problem

tan((17*π)/12)

Solution

1. Identify the angle in terms of more familiar angles. We can split (17*π)/12 into the sum of two angles from the unit circle: (17*π)/12=(14*π)/12+(3*π)/12 which simplifies to (7*π)/6+π/4

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

3. Substitute the values A=(7*π)/6 and B=π/4 into the formula.

4. Evaluate the trigonometric functions at these specific angles: tan((7*π)/6)=√(,3)/3 and tan(π/4)=1

5. Simplify the resulting expression:

tan((17*π)/12)=(√(,3)/3+1)/(1−√(,3)/3*(1))

6. Multiply the numerator and denominator by 3 to clear the fractions:

tan((17*π)/12)=(√(,3)+3)/(3−√(,3))

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

tan((17*π)/12)=((√(,3)+3)*(3+√(,3)))/((3−√(,3))*(3+√(,3)))

8. Expand and simplify the terms:

tan((17*π)/12)=(3√(,3)+3+9+3√(,3))/(9−3)

tan((17*π)/12)=(12+6√(,3))/6

tan((17*π)/12)=2+√(,3)

Final Answer

tan((17*π)/12)=2+√(,3)

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