Find the Exact Value tan(pi/10)

Problem

tan(π/10)

Solution

1. Identify the angle in degrees. Since π radians is 180^∘ the angle is 180^∘/10=18^∘

2. Use the relationship between sine and cosine for 18^∘ We know that sin(18^∘)=(√(,5)−1)/4

3. Determine the value of cos(18^∘) using the Pythagorean identity cos^2(θ)+sin^2(θ)=1

cos(18^∘)=√(,1−sin^2(18^∘))

cos(18^∘)=√(,1−((√(,5)−1)/4)^2)

cos(18^∘)=√(,1−(5−2√(,5)+1)/16)

cos(18^∘)=√(,(16−(6−2√(,5)))/16)

cos(18^∘)=√(,10+2√(,5))/4

4. Apply the definition of tangent, which is tan(θ)=sin(θ)/cos(θ)

tan(π/10)=(√(,5)−1)/4/√(,10+2√(,5))/4

tan(π/10)=(√(,5)−1)/√(,10+2√(,5))

5. Simplify the expression by rationalizing or rewriting the radical. A common simplified form involves multiplying the numerator and denominator by √(,10−2√(,5))

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

tan(π/10)=√(,(6−2√(,5))/(10+2√(,5)))

tan(π/10)=√(,1−2/√(,5))

tan(π/10)=√(,25−10√(,5))/5

Final Answer

tan(π/10)=√(,25−10√(,5))/5

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