Find the Value Using the Unit Circle tan((7pi)/12)

Problem

tan((7*π)/12)

Solution

1. Identify the angle (7*π)/12 as a sum of two standard angles from the unit circle.

(7*π)/12=(3*π)/12+(4*π)/12

2. Simplify the fractions to express the angle in terms of π/4 and π/3

(7*π)/12=π/4+π/3

3. Apply the tangent sum formula tan(A+B)=(tan(A)+tan(B))/(1−tan(A)*tan(B))

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

4. Substitute the known values from the unit circle: tan(π/4)=1 and tan(π/3)=√(,3)

tan((7*π)/12)=(1+√(,3))/(1−(1)*(√(,3)))

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

tan((7*π)/12)=((1+√(,3))*(1+√(,3)))/((1−√(,3))*(1+√(,3)))

6. Expand the numerator and denominator.

tan((7*π)/12)=(1+2√(,3)+3)/(1−3)

7. Simplify the resulting expression.

tan((7*π)/12)=(4+2√(,3))/(−2)

8. Divide each term by −2

tan((7*π)/12)=−2−√(,3)

Final Answer

tan((7*π)/12)=−2−√(,3)

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