Find the Exact Value tan(15 degrees )

Problem

tan(15^∘)

Solution

1. Identify a way to express 15^∘ using angles with known trigonometric values. We can use the difference of 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

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 fraction.

tan(15^∘)=(12−6√(,3))/6

tan(15^∘)=2−√(,3)

Final Answer

tan(15^∘)=2−√(,3)

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