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 sin2(θ)+cos2(θ)=1 to find sin(36)

sin(36)=√(,1−cos2(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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