Find the Exact Value tan(9)

Problem

tan(9^∘)

Solution

1. Identify the angle as 9^∘ and recognize that it can be expressed using half-angle or sum/difference formulas with known angles like 45^∘ and 36^∘ or 54^∘

2. Use the identity tan(θ)=(1−cos(2*θ))/sin(2*θ) where θ=9^∘ so 2*θ=18^∘

3. Recall the exact values for sin(18^∘) and cos(18^∘)

sin(18^∘)=(√(,5)−1)/4

cos(18^∘)=√(,10+2√(,5))/4

4. Substitute these values into the tangent half-angle formula.

tan(9^∘)=(1−√(,10+2√(,5))/4)/(√(,5)−1)/4

5. Simplify the complex fraction by multiplying the numerator and denominator by 4

tan(9^∘)=(4−√(,10+2√(,5)))/(√(,5)−1)

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

tan(9^∘)=((4−√(,10+2√(,5)))*(√(,5)+1))/((√(,5)−1)*(√(,5)+1))

7. Expand the numerator and simplify the denominator, which is $5 - 1 = 4$.

tan(9^∘)=(4√(,5)+4−√(,5)√(,10+2√(,5))−√(,10+2√(,5)))/4

8. Simplify the term √(,5)√(,10+2√(,5)) as √(,50+10√(,5))

tan(9^∘)=√(,5)+1−√(,5+2√(,5))

Final Answer

tan(9^∘)=√(,5)+1−√(,5+2√(,5))

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