Find the Exact Value cot(36 degrees )

Problem

cot(36^∘)

Solution

1. Identify the relationship between the angle 36^∘ and the golden ratio ϕ The value of cos(36^∘) is known to be (1+√(,5))/4

2. Use the Pythagorean identity sin^2(θ)+cos^2(θ)=1 to find sin(36^∘)

sin(36^∘)=√(,1−cos^2(36^∘))

sin(36^∘)=√(,1−((1+√(,5))/4)^2)

sin(36^∘)=√(,1−(1+2√(,5)+5)/16)

sin(36^∘)=√(,(16−6−2√(,5))/16)

sin(36^∘)=√(,10−2√(,5))/4

3. Apply the definition of the cotangent function, which is the ratio of cosine to sine.

cot(36^∘)=cos(36^∘)/sin(36^∘)

cot(36^∘)=(1+√(,5))/4/√(,10−2√(,5))/4

cot(36^∘)=(1+√(,5))/√(,10−2√(,5))

4. Simplify the expression by rationalizing or rewriting. To express it in a standard radical form, we can square the numerator inside a square root.

cot(36^∘)=√(,((1+√(,5))^2)/(10−2√(,5)))

cot(36^∘)=√(,(6+2√(,5))/(10−2√(,5)))

cot(36^∘)=√(,((6+2√(,5))*(10+2√(,5)))/((10−2√(,5))*(10+2√(,5))))

cot(36^∘)=√(,(60+12√(,5)+20√(,5)+20)/(100−20))

cot(36^∘)=√(,(80+32√(,5))/80)

cot(36^∘)=√(,1+(2√(,5))/5)

cot(36^∘)=√(,1+2/√(,5))

Final Answer

cot(36^∘)=√(,1+2/√(,5))

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