Find the Exact Value tan(36)

Problem

tan(36^∘)

Solution

1. Identify the relationship between 36^∘ and 72^∘ We know that 2×36^∘=72^∘ and 3×36^∘=108^∘ Since sin(108^∘)=sin(180^∘−108^∘)=sin(72^∘) we can use the triple angle and double angle formulas for sine.

2. Determine the value of cos(36^∘) using the golden ratio ϕ=(1+√(,5))/2 The exact value is known to be:

cos(36^∘)=(1+√(,5))/4

3. Calculate the value of sin(36^∘) using the Pythagorean identity sin^2(θ)+cos^2(θ)=1

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

4. Apply the definition of the tangent function, which is the ratio of sine to cosine.

tan(36^∘)=sin(36^∘)/cos(36^∘)

tan(36^∘)=√(,10−2√(,5))/4/(1+√(,5))/4

tan(36^∘)=√(,10−2√(,5))/(1+√(,5))

5. Rationalize the denominator by multiplying the numerator and denominator by √(,5)−1

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

tan(36^∘)=√(,(10−2√(,5))*(√(,5)−1)^2)/(5−1)

tan(36^∘)=√(,(10−2√(,5))*(5−2√(,5)+1))/4

tan(36^∘)=√(,(10−2√(,5))*(6−2√(,5)))/4

tan(36^∘)=√(,60−20√(,5)−12√(,5)+20)/4

tan(36^∘)=√(,80−32√(,5))/4

6. Simplify the radical by factoring out 16 from the expression inside the square root.

tan(36^∘)=√(,16*(5−2√(,5)))/4

tan(36^∘)=(4√(,5−2√(,5)))/4

tan(36^∘)=√(,5−2√(,5))

Final Answer

tan(36^∘)=√(,5−2√(,5))

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