Find the Exact Value sin(72)

Problem

sin(72^∘)

Solution

1. Identify the angle as part of a 36^∘−72^∘−72^∘ isosceles triangle, which is related to the golden ratio ϕ=(1+√(,5))/2

2. Use the identity for the cosine of 36^∘ which is known to be cos(36^∘)=(1+√(,5))/4

3. Apply the cofunction identity sin(72^∘)=cos(18^∘) and the half-angle formula or power-reduction identity. Alternatively, use the identity sin(72^∘)=√(,1−cos^2(72^∘))

4. Determine cos(72^∘) using the double-angle formula cos(72^∘)=2*cos^2(36^∘)−1 or by using the property cos(72^∘)=(√(,5)−1)/4

5. Substitute the value of cos(72^∘) into the Pythagorean identity sin(72^∘)=√(,1−cos^2(72^∘))

6. Simplify the expression inside the square root.

sin(72^∘)=√(,1−((√(,5)−1)/4)^2)

sin(72^∘)=√(,1−(5−2√(,5)+1)/16)

sin(72^∘)=√(,(16−(6−2√(,5)))/16)

sin(72^∘)=√(,(10+2√(,5))/16)

sin(72^∘)=√(,10+2√(,5))/4

Final Answer

sin(72^∘)=√(,10+2√(,5))/4

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