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

Problem

tan(π/6−π/4)

Solution

1. Identify the appropriate trigonometric identity for the tangent of a difference, 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 identity.

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

3. Evaluate the basic tangent values using the unit circle, where 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 products in the numerator and denominator.

tan(π/6−π/4)=(3√(,3)−3−9+3√(,3))/(9−3)

8. Combine like terms and simplify the fraction.

tan(π/6−π/4)=(6√(,3)−12)/6

tan(π/6−π/4)=√(,3)−2

Final Answer

tan(π/6−π/4)=√(,3)−2

Want more problems? Check here!