Find the Exact Value sin(9)

Problem

sin(9^∘)

Solution

1. Identify the angle as 9^∘ which can be expressed as half of 18^∘

2. Recall the exact value of sin(18^∘) which is derived from a regular pentagon or the golden ratio.

sin(18^∘)=(√(,5)−1)/4

3. Apply the half-angle formula for sine, sin(θ/2)=√(,(1−cos(θ))/2) noting that 9^∘ is in the first quadrant so the result is positive.

sin(9^∘)=√(,(1−cos(18^∘))/2)

4. Determine the value of cos(18^∘) using the identity cos(θ)=√(,1−sin^2(θ))

cos(18^∘)=√(,1−((√(,5)−1)/4)^2)

cos(18^∘)=√(,1−(5−2√(,5)+1)/16)

cos(18^∘)=√(,(16−6+2√(,5))/16)

cos(18^∘)=√(,10+2√(,5))/4

5. Substitute the value of cos(18^∘) back into the half-angle formula.

sin(9^∘)=√(,(1−√(,10+2√(,5))/4)/2)

6. Simplify the expression by finding a common denominator inside the square root.

sin(9^∘)=√(,(4−√(,10+2√(,5)))/8)

7. Rationalize the denominator by multiplying the numerator and denominator inside the root by 2

sin(9^∘)=√(,(8−2√(,10+2√(,5)))/16)

sin(9^∘)=√(,8−2√(,10+2√(,5)))/4

Final Answer

sin(9^∘)=√(,8−2√(,10+2√(,5)))/4

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