Find the Exact Value tan(pi/6+pi/4)

Problem

tan(π/6+π/4)

Solution

1. Identify the sum formula for tangent, which is tan(A+B)=(tan(A)+tan(B))/(1−tan(A)*tan(B))

2. Substitute the values A=π/6 and B=π/4 into the formula.

tan(π/6+π/4)=(tan(π/6)+tan(π/4))/(1−tan(π/6)*tan(π/4))

3. Evaluate the trigonometric functions using the unit circle: tan(π/6)=√(,3)/3 and tan(π/4)=1

tan(π/6+π/4)=(√(,3)/3+1)/(1−(√(,3)/3)*(1))

4. Simplify the numerator and denominator by finding a common denominator of 3

tan(π/6+π/4)=(√(,3)+3)/3/(3−√(,3))/3

5. Cancel the common denominator of 3 from the complex fraction.

tan(π/6+π/4)=(3+√(,3))/(3−√(,3))

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

tan(π/6+π/4)=((3+√(,3))*(3+√(,3)))/((3−√(,3))*(3+√(,3)))

7. Expand the terms in the numerator and denominator.

tan(π/6+π/4)=(9+6√(,3)+3)/(9−3)

8. Simplify the resulting expression.

tan(π/6+π/4)=(12+6√(,3))/6

9. Divide each term in the numerator by 6

tan(π/6+π/4)=2+√(,3)

Final Answer

tan(π/6+π/4)=2+√(,3)

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