Find the Exact Value csc(9)

Problem

csc(9^∘)

Solution

1. Identify the expression as the cosecant of 9 degrees. Since csc(θ)=1/sin(θ) we must first find the value of sin(9^∘)

2. Apply the half-angle formula for sine, sin(θ/2)=√(,(1−cos(θ))/2) using θ=18^∘

3. Substitute the known value cos(18^∘)=√(,10+2√(,5))/4 into the half-angle formula.

4. Simplify the expression for sin(9^∘)

sin(9^∘)=√(,(1−√(,10+2√(,5))/4)/2)

sin(9^∘)=√(,(4−√(,10+2√(,5)))/8)

sin(9^∘)=√(,8−2√(,10+2√(,5)))/4

5. Invert the sine value to find the cosecant:

csc(9^∘)=4/√(,8−2√(,10+2√(,5)))

6. Rationalize or simplify the radical expression. An alternative exact form derived from the difference of angles 54^∘−45^∘ or 45^∘−36^∘ is often expressed as:

csc(9^∘)=√(,2)+√(,10)+√(,5+2√(,5))

Final Answer

csc(9^∘)=√(,2)+√(,10)+√(,5+2√(,5))

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