Find the Exact Value tan(165 degrees )

Problem

tan(165)

Solution

1. Identify the angle as a sum of two special angles from the unit circle.

tan(165)=tan(120+45)

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

tan(120+45)=(tan(120)+tan(45))/(1−tan(120)*tan(45))

3. Substitute the known values tan(120)=−√(,3) and tan(45)=1 into the expression.

tan(165)=(−√(,3)+1)/(1−(−√(,3))*(1))

4. Simplify the expression by rearranging the terms in the numerator and denominator.

tan(165)=(1−√(,3))/(1+√(,3))

5. Rationalize the denominator by multiplying the numerator and denominator by the conjugate 1−√(,3)

tan(165)=((1−√(,3))*(1−√(,3)))/((1+√(,3))*(1−√(,3)))

6. Expand the numerator and denominator.

tan(165)=(1−2√(,3)+3)/(1−3)

7. Combine like terms and simplify the fraction.

tan(165)=(4−2√(,3))/(−2)

8. Divide each term in the numerator by −2

tan(165)=−2+√(,3)

Final Answer

tan(165)=√(,3)−2

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