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 sin2(θ)+cos2(θ)=1

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

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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