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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