Find the Exact Value tan(15)

Problem

tan(15)

Solution

1. Identify the angle as a difference of two special angles from the unit circle, such as 45 and 30

tan(15)=tan(45−30)

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

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

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

tan(15)=(1−√(,3)/3)/(1+(1)*(√(,3)/3))

4. Simplify the complex fraction by multiplying the numerator and the denominator by 3

tan(15)=(3−√(,3))/(3+√(,3))

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

tan(15)=((3−√(,3))*(3−√(,3)))/((3+√(,3))*(3−√(,3)))

6. Expand the numerator and denominator.

tan(15)=(9−6√(,3)+3)/(9−3)

7. Combine like terms and simplify the resulting fraction.

tan(15)=(12−6√(,3))/6

8. Divide each term in the numerator by 6

tan(15)=2−√(,3)

Final Answer

tan(15)=2−√(,3)

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