Find the Exact Value tan(pi/3+pi/4)

Problem

tan(π/3+π/4)

Solution

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

2. Substitute the values A=π/3 and B=π/4 into the formula.

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

3. Evaluate the trigonometric values for the special angles tan(π/3)=√(,3) and tan(π/4)=1

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

4. Simplify the expression to obtain the ratio.

tan(π/3+π/4)=(√(,3)+1)/(1−√(,3))

5. Rationalize the denominator by multiplying the numerator and denominator by the conjugate 1+√(,3)

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

6. Expand the numerator and denominator.

tan(π/3+π/4)=(3+2√(,3)+1)/(1−3)

7. Simplify the resulting fraction.

tan(π/3+π/4)=(4+2√(,3))/(−2)

8. Divide each term in the numerator by −2

tan(π/3+π/4)=−2−√(,3)

Final Answer

tan(π/3+π/4)=−2−√(,3)

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