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

Want more problems? Check here!