Find the Exact Value tan(105)

Problem

tan(105)

Solution

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

tan(105)=tan(60+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(60+45)=(tan(60)+tan(45))/(1−tan(60)*tan(45))

3. Substitute the known values tan(60)=√(,3) and tan(45)=1

tan(105)=(√(,3)+1)/(1−(√(,3))*(1))

4. Simplify the expression.

tan(105)=(√(,3)+1)/(1−√(,3))

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

tan(105)=((√(,3)+1)*(1+√(,3)))/((1−√(,3))*(1+√(,3)))

6. Expand the numerator and denominator.

tan(105)=(√(,3)+3+1+√(,3))/(1−3)

7. Combine like terms.

tan(105)=(4+2√(,3))/(−2)

8. Divide each term by −2

tan(105)=−2−√(,3)

Final Answer

tan(105)=−2−√(,3)

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