Find the Exact Value cos(72)

Problem

cos(72^∘)

Solution

1. Identify the angle as 72^∘ which is equivalent to (2*π)/5 radians. This angle is related to the geometry of a regular pentagon.

2. Use the relationship between the golden ratio ϕ=(1+√(,5))/2 and the trigonometric functions of angles related to the pentagon.

3. Apply the formula for cos(72^∘) which is known to be (√(,5)−1)/4 This can be derived from the double angle formula cos(72^∘)=sin(18^∘)

4. Verify using the identity sin(18^∘)=(√(,5)−1)/4 Since 72^∘ and 18^∘ are complementary angles, cos(72^∘)=sin(90^∘−72^∘)=sin(18^∘)

Final Answer

cos(72^∘)=(√(,5)−1)/4

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