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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