Find the Exact Value tan(-75)

Problem

tan(−75)

Solution

1. Apply the odd function property of the tangent function, which states that tan(−θ)=−tan(θ)

tan(−75)=−tan(75)

2. Rewrite the angle as a sum of two special angles from the unit circle for which the exact values are known.

75=45+30

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

−tan(75)=−(tan(45)+tan(30))/(1−tan(45)*tan(30))

4. Substitute the known values tan(45)=1 and tan(30)=√(,3)/3

−tan(75)=−(1+√(,3)/3)/(1−(1)*(√(,3)/3))

5. Simplify the fraction by multiplying the numerator and the denominator by 3

−tan(75)=−(3+√(,3))/(3−√(,3))

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

−tan(75)=−((3+√(,3))*(3+√(,3)))/((3−√(,3))*(3+√(,3)))

7. Expand and simplify the expression.

−tan(75)=−(9+6√(,3)+3)/(9−3)

−tan(75)=−(12+6√(,3))/6

−tan(75)=−(2+√(,3))

Final Answer

tan(−75)=−2−√(,3)

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