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−sin2(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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