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

Problem

tan(π/4−π/3)

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/4 and B=π/3 into the identity.

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

3. Evaluate the basic tangent values using the unit circle: tan(π/4)=1 and tan(π/3)=√(,3)

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

4. Simplify the expression into a single fraction.

tan(π/4−π/3)=(1−√(,3))/(1+√(,3))

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

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

6. Expand the numerator and denominator.

tan(π/4−π/3)=(1−2√(,3)+3)/(1−3)

7. Combine like terms and simplify the fraction.

tan(π/4−π/3)=(4−2√(,3))/(−2)

tan(π/4−π/3)=−2+√(,3)

Final Answer

tan(π/4−π/3)=√(,3)−2

Want more problems? Check here!