Expand Using Sum/Difference Formulas tan(165)

Problem

tan(165^∘)

Solution

1. Identify two angles with known trigonometric values that sum or subtract to 165^∘ We can use 120^∘ and 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))

3. Substitute A=120^∘ and B=45^∘ into the formula.

tan(120^∘+45^∘)=(tan(120^∘)+tan(45^∘))/(1−tan(120^∘)*tan(45^∘))

4. Evaluate the tangent values for the specific angles. We know tan(120^∘)=−√(,3) and tan(45^∘)=1

tan(165^∘)=(−√(,3)+1)/(1−(−√(,3))*(1))

5. Simplify the expression.

tan(165^∘)=(1−√(,3))/(1+√(,3))

6. Rationalize the denominator by multiplying the numerator and denominator by the conjugate 1−√(,3)

tan(165^∘)=((1−√(,3))*(1−√(,3)))/((1+√(,3))*(1−√(,3)))

7. Expand the numerator and denominator.

tan(165^∘)=(1−2√(,3)+3)/(1−3)

8. Simplify the final fraction.

tan(165^∘)=(4−2√(,3))/(−2)

tan(165^∘)=−2+√(,3)

Final Answer

tan(165^∘)=√(,3)−2

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