Find the Exact Value cot(81)

Problem

cot(81)

Solution

1. Identify the angle as 81 and recognize that it can be expressed as 90−9

2. Apply the cofunction identity cot(90−θ)=tan(θ)

cot(81)=tan(9)

3. Use the half-angle formula for tangent, tan(θ/2)=(1−cos(θ))/sin(θ) where θ=18

tan(9)=(1−cos(18))/sin(18)

4. Substitute the known exact values for sin(18) and cos(18)

sin(18)=(√(,5)−1)/4

cos(18)=√(,10+2√(,5))/4

5. Plug these values into the expression for tan(9)

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

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

tan(9)=(4−√(,10+2√(,5)))/(√(,5)−1)

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

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

8. Expand the numerator and simplify the denominator.

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

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

tan(9)=(4√(,5)+4−√(,50+10√(,5))−√(,10+2√(,5)))/4

10. Divide each term by 4 to reach the final form.

tan(9)=√(,5)+1−(√(,50+10√(,5))+√(,10+2√(,5)))/4

Final Answer

cot(81)=(4√(,5)+4−√(,50+10√(,5))−√(,10+2√(,5)))/4

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