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−cos2(72))

4. Determine cos(72) using the double-angle formula cos(72)=2*cos2(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−cos2(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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