Find the Exact Value cos((4pi)/5)

Problem

cos((4*π)/5)

Solution

1. Identify the angle in terms of a more familiar reference angle. Since (4*π)/5=π−π/5 we can use the supplementary angle identity cos(π−θ)=−cos(θ)

2. Apply the identity to rewrite the expression.

cos((4*π)/5)=−cos(π/5)

3. Recall the exact value for cos(π/5) which is cos(36) This value is derived from the properties of a regular pentagon or the golden ratio ϕ=(1+√(,5))/2

cos(π/5)=(1+√(,5))/4

4. Substitute this value back into the expression from step 2.

cos((4*π)/5)=−((1+√(,5))/4)

5. Simplify the signs to find the final exact value.

cos((4*π)/5)=(−1−√(,5))/4

Final Answer

cos((4*π)/5)=(−1−√(,5))/4

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