Find the Exact Value cot(pi/10)

Problem

cot(π/10)

Solution

1. Identify the angle in degrees to recognize its properties. Since π radians is 180^∘ the angle is 180^∘/10=18^∘

2. Relate the cotangent function to the sine and cosine functions using the identity cot(θ)=cos(θ)/sin(θ)

3. Determine the exact value of sin(18^∘) Using the properties of a golden triangle or the multiple-angle formula 3*θ=90^∘−2*θ we find sin(18^∘)=(√(,5)−1)/4

4. Determine the exact value of cos(18^∘) using the Pythagorean identity cos(θ)=√(,1−sin^2(θ)) Substituting the sine value gives cos(18^∘)=√(,10+2√(,5))/4

5. Substitute these values into the cotangent ratio:

cot(π/10)=√(,10+2√(,5))/4/(√(,5)−1)/4

6. Simplify the fraction by canceling the denominators and rationalizing the expression:

cot(π/10)=√(,10+2√(,5))/(√(,5)−1)

7. Multiply the numerator and denominator by the conjugate √(,5)+1 and simplify the radical expression to reach the standard form.

Final Answer

cot(π/10)=√(,5+2√(,5))

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