Expand Using Sum/Difference Formulas tan(15)

Problem

tan(15^∘)

Solution

1. Identify two common angles from the unit circle whose difference is 15^∘ We can use 45^∘ and 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))

3. Substitute A=45^∘ and B=30^∘ into the formula.

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

4. Evaluate the trigonometric values: tan(45^∘)=1 and tan(30^∘)=√(,3)/3

tan(15^∘)=(1−√(,3)/3)/(1+(1)*(√(,3)/3))

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

tan(15^∘)=(3−√(,3))/(3+√(,3))

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

tan(15^∘)=((3−√(,3))^2)/((3+√(,3))*(3−√(,3)))

7. Expand and simplify the terms.

tan(15^∘)=(9−6√(,3)+3)/(9−3)

tan(15^∘)=(12−6√(,3))/6

tan(15^∘)=2−√(,3)

Final Answer

tan(15^∘)=2−√(,3)

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