Find the Exact Value tan(pi/5)

Problem

tan(π/5)

Solution

1. Identify the angle as 36 To find the exact value of tan(π/5) we use the relationship tan(θ)=sin(θ)/cos(θ)

2. Recall the exact values for the sine and cosine of π/5 These are derived from properties of a regular pentagon or the golden ratio ϕ=(1+√(,5))/2

3. Substitute the known value cos(π/5)=(1+√(,5))/4

4. Calculate the sine value using the identity sin2(θ)+cos2(θ)=1

sin(π/5)=√(,1−((1+√(,5))/4)2)

5. Simplify the expression for sine.

sin(π/5)=√(,1−(1+2√(,5)+5)/16)

sin(π/5)=√(,(16−6−2√(,5))/16)

sin(π/5)=√(,10−2√(,5))/4

6. Divide the sine by the cosine to find the tangent.

tan(π/5)=√(,10−2√(,5))/4/(1+√(,5))/4

tan(π/5)=√(,10−2√(,5))/(1+√(,5))

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

tan(π/5)=(√(,10−2√(,5))*(√(,5)−1))/((√(,5)+1)*(√(,5)−1))

tan(π/5)=√(,(10−2√(,5))*(√(,5)−1)2)/(5−1)

tan(π/5)=√(,(10−2√(,5))*(5−2√(,5)+1))/4

tan(π/5)=√(,(10−2√(,5))*(6−2√(,5)))/4

tan(π/5)=√(,60−20√(,5)−12√(,5)+20)/4

tan(π/5)=√(,80−32√(,5))/4

8. Simplify the radical by factoring out 16

tan(π/5)=√(,16*(5−2√(,5)))/4

tan(π/5)=√(,5−2√(,5))

Final Answer

tan(π/5)=√(,5−2√(,5))

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