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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