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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