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−sin2(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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