Find the Exact Value cot(pi/5)

Problem

cot(π/5)

Solution

1. Identify the angle in degrees. Since π radians equals 180 the angle is 180/5=36=36

2. Use the relationship between the cotangent, cosine, and sine functions.

cot(π/5)=cos(36)/sin(36)

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

cos(36)=(1+√(,5))/4

4. Determine the value of sin(36) using the Pythagorean identity sin2(θ)+cos2(θ)=1

sin(36)=√(,1−((1+√(,5))/4)2)

5. Simplify the expression for sin(36)

sin(36)=√(,1−(1+2√(,5)+5)/16)

sin(36)=√(,(16−6−2√(,5))/16)

sin(36)=√(,10−2√(,5))/4

6. Substitute these values back into the cotangent formula.

cot(π/5)=(1+√(,5))/4/√(,10−2√(,5))/4

7. Simplify the fraction by canceling the denominators.

cot(π/5)=(1+√(,5))/√(,10−2√(,5))

8. Rationalize or simplify the radical expression if necessary. By squaring the expression and simplifying, or using half-angle identities, the expression can be written in a standard radical form.

cot(π/5)=√(,((1+√(,5))2)/(10−2√(,5)))

cot(π/5)=√(,(6+2√(,5))/(10−2√(,5)))

cot(π/5)=√(,1+2/√(,5))

cot(π/5)=√(,(5+2√(,5))/5)

Final Answer

cot(π/5)=√(,1+(2√(,5))/5)

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