Find the Exact Value tan(72)

Problem

tan(72^∘)

Solution

1. Identify the angle as a multiple of 18^∘ and use the relationship 72^∘=90^∘−18^∘

2. Apply the cofunction identity tan(90^∘−θ)=cot(θ) to rewrite the expression.

tan(72^∘)=cot(18^∘)

3. Recall the exact value of sin(18^∘) using the golden ratio ϕ

sin(18^∘)=(√(,5)−1)/4

4. Calculate cos(18^∘) using the Pythagorean identity cos(θ)=√(,1−sin^2(θ))

cos(18^∘)=√(,1−((√(,5)−1)/4)^2)

cos(18^∘)=√(,10+2√(,5))/4

5. Substitute these values into the definition cot(18^∘)=cos(18^∘)/sin(18^∘)

cot(18^∘)=√(,10+2√(,5))/(√(,5)−1)

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

cot(18^∘)=(√(,10+2√(,5))*(√(,5)+1))/4

7. Simplify the expression inside the square root or use the alternative radical form.

tan(72^∘)=√(,5+2√(,5))

Final Answer

tan(72^∘)=√(,5+2√(,5))

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